{"id":2539,"date":"2026-01-02T11:29:22","date_gmt":"2026-01-02T05:59:22","guid":{"rendered":"https:\/\/www.manabadi.co.in\/boards\/?p=2539"},"modified":"2026-01-03T11:33:50","modified_gmt":"2026-01-03T06:03:50","slug":"ts-inter-1st-year-maths-1a-solutions-cp-2-mathematical-induction","status":"publish","type":"post","link":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-solutions-cp-2-mathematical-induction\/","title":{"rendered":"TS Inter 1st Year Maths 1A Solutions Chapter 2 Mathematical Induction Ex 2(a),TG Inter 1st Year Maths 1A Mathematical Induction Solutions Exercise 2(a)Using Mathematical Induction prove each of the following statement for all n \u2208 N."},"content":{"rendered":"\n

Question 1.
12<\/sup> + 22<\/sup>+ 32<\/sup> + \u2026\u2026\u2026\u2026. + n2<\/sup> = n(n+1)(2n+1)\/6<\/h3>\n\n\n\n

Answer:
Let S(n) be the given statement
12<\/sup> + 22<\/sup> + 32<\/sup> + \u2026\u2026\u2026\u2026. + n2<\/sup> = n(n+1)(2n+1)\/6
Since 12<\/sup> = 1(1+1)(2+1)\/6
\u21d2 1 = 1; the statement is true for n = 1
Suppose the statement is true for n = k then
(12<\/sup> + 22<\/sup> + 32<\/sup> + \u2026\u2026\u2026\u2026.. + k2<\/sup>) + k(k+1)(2k+1)\/6
We have to prove that the statement is true for n = k + 1 also then
(12<\/sup> + 22<\/sup> + 32<\/sup> + \u2026\u2026\u2026\u2026\u2026 + k2<\/sup>) + (k + 1)2<\/sup>

\u2234 The statement is true for n = k + 1 also.
\u2234 By the principle of Finite Mathematical Induction S(n) is true for all n \u2208 N.
i.e., 12<\/sup> + 22<\/sup> + 32<\/sup> + \u2026\u2026\u2026.. + n2<\/sup> = n(n+1)(2n+1)\/6, \u2200 n \u2208 N<\/p>\n\n\n\n

Question 2. 2.3 + 3.4 + 4.5 + \u2026\u2026\u2026\u2026\u2026\u2026. upto n terms = n(n2<\/sup>+6n+11)\/3 (March 13, May 06)<\/h3>\n\n\n\n

Answer:
Let S(n) be the statement.
The nth term of 2.3 + 3.4 + 4.5 + \u2026\u2026\u2026\u2026\u2026 is (n + 1) (n + 2)
\u2234 2 . 3 + 3 . 4 + 4 . 5 + \u2026\u2026\u2026\u2026\u2026.. + (n + 1) (n + 2)
= n(n2<\/sup>+6n+11)\/3
Now S(1) = 2 . 3 = 1(12<\/sup>+6+11)\/3 = 6
\u2234 The statement is true for n = 1.
Suppose that the statement is true for n = k, then 2.3 + 3.4 + 4.5 + \u2026\u2026\u2026\u2026\u2026.. + (k + 1) (k + 2)
= k(k2<\/sup>+6k+11)\/3
Adding (k + 1) th term of L.H.S both sides
S(k + 1) = 2.3 + 3.4 + 4.5 + \u2026\u2026\u2026\u2026\u2026. + k (k + 1) (k + 2) + (k + 2) (k + 3)

\u2234 S(k + 1) = 1\/3 (k + 1) [k2<\/sup> + 2k + 1 + 6 (k + 1) + 11]
= 1\/3 (k + 1) [(k + 1)2 + 6(k + 1) + 11]
\u2234 The statement is true for n = k + 1
So by the principle of Finite Mathematical Induction S(n) is true \u2200 n \u2208 N
\u2234 2 . 3 + 3 . 4 + 4 . 5 + \u2026\u2026\u2026\u2026\u2026. + (n + 1) (n + 2) = n(n2<\/sup>+6n+11)\/3<\/p>\n\n\n\n

Question 3. 1\/1\u22c53+1\/3\u22c55+1\/5\u22c57 + \u2026\u2026\u2026\u2026\u2026. + 1\/(2n\u22121)(2n+1) = n2n\/+1 (May 2014)<\/h3>\n\n\n\n

Answer:
Let Sn be the statement
1\/1\u22c53+1\/3\u22c55+1\/5\u22c57 + \u2026\u2026\u2026\u2026\u2026. + 1\/(2n\u22121)(2n+1)
Then S(1) = 1\/1\u22c53=1\/2(1)+1=1\/3
\u2234 S(1) is true.
Suppose the statement is true for n = k, then
S(K) = 1\/1\u22c53+1\/3\u22c55+1\/5\u22c57+\u2026.+1(2k\u22121)(2k+1) = k\/2k+1
We have to show that the statement is true for n = k + 1 also,

The statement S(n) is true for n = k + 1
\u2234 By the principle of Mathematical Induction S(n) is true for all n \u2208 N.
\u2234 1\/1\u22c53+1\/3\u22c55+1\/5\u22c57 + \u2026\u2026\u2026\u2026\u2026. + 1\/(2n\u22121)(2n+1) = n\/2n+1<\/p>\n\n\n\n

Question 4. 43<\/sup> + 83<\/sup> + 123<\/sup> + \u2026\u2026\u2026\u2026 + upto n terms = 16n2<\/sup> (n + 1)2<\/sup><\/h3>\n\n\n\n

Answer:
4, 8, 12, are in A.P. and nth term of A.P. is = a + (n \u2013 1) d
= 4 + (n \u2013 1) 4 = 4n
Let S(n) be the statement
43<\/sup> + 83<\/sup> + 123<\/sup> + \u2026\u2026\u2026\u2026\u2026\u2026 + (4n)3<\/sup> = 16n2<\/sup> (n + 1)2<\/sup>
Let n = 1, then
S(1) = 43<\/sup> = 16 (1 + 1)2<\/sup> = 64
\u2234 The statement is true for n = 1 also.
Suppose the statement is true for n = k then
43<\/sup> + 83<\/sup> + 123<\/sup>+ \u2026\u2026\u2026\u2026\u2026\u2026 + (4k)3<\/sup> = 16k2<\/sup> (k + 1)2<\/sup>
We have to prove that the result is true for n = k + 1 also. Adding (k + 1) th term
= [4 (k + 1)]3<\/sup> = [4k + 4]3<\/sup> both sides
43<\/sup> + 83<\/sup> + 123<\/sup> + \u2026\u2026\u2026\u2026\u2026\u2026.. + (4k)3<\/sup> + (4k + 4)3<\/sup>
= 16k3<\/sup> (k + 1)2 + [4 (k + 1)]3<\/sup>
= 16 (k + 1)2<\/sup> [k2<\/sup> + 4k + 4]
= 16 (k + 1)2<\/sup> (k + 2)2<\/sup>
= 16 (k + 1)2<\/sup> [(k + 1) + 1]2<\/sup>
Hence the result is true for n = k + 1.
\u2234 By the principle of Mathematical Induction S(n) is true \u2200 n \u2208 N.
\u2234 43<\/sup> + 83<\/sup> + 123<\/sup> + \u2026\u2026\u2026\u2026\u2026\u2026.. + (4n)3<\/sup> = 16n2 (n + 1)2<\/p>\n\n\n\n

Question 5.
a + (a + d) + (a + 2d) + \u2026\u2026\u2026\u2026\u2026\u2026 upto n terms = n\/2 [2a + (n \u2013 1) d]<\/h3>\n\n\n\n

Answer:
Let S(n) be the statement
a + (a + d) + (a + 2d) + + [a + (n \u2013 1) d] = n\/2 [2a + (n \u2013 1) d]
Now S(1) is a = 1\/2 [2a + 0 (d)] = a
\u2234 S(1) is true.
Assume that the statement is true for n = k
\u2234 S(k) = a + (a + d) + (a + 2d) + \u2026\u2026\u2026\u2026\u2026\u2026.. + [a + (k \u2013 1) d]
= k\/2 [2a + (k \u2013 1) d]
We have to prove that the statement is true for n = k + 1 also.
Adding a + kd both sides
a + (a + d) + \u2026\u2026\u2026\u2026\u2026. + [a + (k \u2013 1) d] + [a + kd]

\u2234 The statement is true for n = k + 1 also
\u2234 By the principle of Mathematical Induction.
S(n) is true for all n \u2208 N
\u2234 a + (a + d) + (a + 2d) + + [a + (n \u2013 1) d] = n\/2 [2a + (n -1) d]
<\/p>\n\n\n\n

Question 6.
a + ar + ar2<\/sup> + \u2026\u2026\u2026\u2026\u2026\u2026\u2026 + n terms = a(rn<\/sup>\u22121)\/r\u22121, r \u2260 1 (March 2011)<\/h3>\n\n\n\n

Answer:
Let S(n) be the statement
a + ar + ar2<\/sup> + \u2026\u2026\u2026\u2026.. + arn \u2013 1 = (rn<\/sup>\u22121)\/r\u22121, r \u2260 1
Then S(1) = a = a(r1<\/sup>\u22121)\/r\u22121 = a
\u2234 The result is true for n = 1
Suppose the statement is true for n = k then
a + ar + ar2<\/sup> + \u2026\u2026\u2026\u2026\u2026 + ar = a(rk<\/sup>\u22121)\/r\u22121, r \u2260 1
We have to prove that the result is true for n = k + 1 also.
Adding ark both sides
(a + ar + ar2<\/sup> + \u2026\u2026\u2026\u2026.. + ark \u2013 1<\/sup> + ark<\/sup>)

\u2234 The statement is true for n = k + 1 also
\u2234 By the principle of Mathematical Induction p(n) is true for all n \u2208 N
a + ar + ar2<\/sup> + \u2026\u2026\u2026\u2026\u2026\u2026\u2026 + n terms = a(rn<\/sup>\u22121)\/r\u22121, r \u2260 1
<\/p>\n\n\n\n

Question 7.
2 + 7 + 12 + \u2026\u2026\u2026. + (5n \u2013 3) = n(5n\u22121)\/2<\/h3>\n\n\n\n

Answer:
Let S(n) be the statement
2 + 7 + 12 + \u2026\u2026\u2026. + (5n \u2013 3) = n(5n\u22121)\/2 = 1(5\u22121)\/2 = 2,
Since S(1) = 2,
S(1) is true.
Suppose the statement is true for n k then
(2 + 7 + 12 + \u2026\u2026\u2026\u2026.. + (5k \u2013 3) = k(5k\u22121)\/2
We have to show that S(n) is true for n = k + 1 also.
Adding (k + 1)th term 5 (k + 1) \u2013 3 = 5k + 2 both sides
[2 + 7 + 12 + \u2026\u2026. + 5k \u2013 3)] + (5k + 2)

\u2234 S(n) is true for n = k + 1 also
\u2234 By the principal of Mathematical Induction
S(n) is true \u2200 n \u2208 N
\u2234 2 + 7 + 12 + \u2026\u2026\u2026\u2026\u2026 + (5n \u2013 3) = n(5n\u22121)\/2.
<\/p>\n\n\n\n

Question 8.
(1 + 3\/1)(1 + 5\/4)(1 + 7\/9)\u2026\u2026\u2026(1 + 2n+1\/n2) = (n + 1)2 (March 2015-A.P)<\/sup><\/h3>\n\n\n\n


Answer:
Let Sn<\/sub> be the statement

\u2234 S(n) is true for n = 1
Suppose Sn is true for n = k then
(1 + 3\/1)(1 + 5\/4)(1 + 7\/9)\u2026\u2026(1 + 2k+1\/k2)
= (k + 1)2 \u2026\u2026\u2026\u2026\u2026\u2026. (1)
We have to prove that the statement is true for n = k + 1 also
(k + 1) th term is

= k2 + 4k + 4
= (k + 2)2
\u2234 S(n) is true for n = k + 1 also
\u2234 By the principal of Mathematical Induction S(n) is true for \u2200 n \u2208 N<\/sup><\/p>\n\n\n\n


Question 9.
(2n + 1) < (n + 3)2
<\/h3>\n\n\n\n

Answer:
Let S(n) be the statement
When n = 1, then 9 < 16 \u2234 S(n) is true for n = 1 Suppose S(n) is true for n = k then(2k + 7) < (k + 3)2 \u2026\u2026\u2026\u2026\u2026.. (1)
We have to prove that the result is true for n = k + 1
i.e., 2(k + 1) + 7 < (k + 4)2
\u2234 2 (k + 1) + 7 = 2k + 2 + 7 = (2k + 7) + 2 < (k + 3)2 + 2 (From (1))
= k2 + 6k + 9 + 2
= k2 + 6k + 11 < (k2 + 6k + 11) + (2k + 5)
= k2 + 8k + 16
= (k + 4)2
\u2234 S(n) is true for n = k + 1 also
By principle of Mathematical Induction
S(n) is true \u2200 n \u2208 N<\/sup><\/p>\n\n\n\n


Question 10.
12 + 22 + \u2026\u2026\u2026\u2026\u2026. + n2 > n3\/3
<\/h3>\n\n\n\n

Answer:
Let S(n) be the statement
When n = 1, then 1 > 1\/3
\u2234 S(n) is true for n = 1
Assume S(n) to be true for n = k then
12 + 22 + \u2026\u2026\u2026\u2026\u2026. + k2 > k3\/3
We have to prove that the result is true for n = k + 1 also.

\u2234 S(n) is true for n = k + 1 also
\u2234 By principle of Mathematical Induction.
S(n) is true \u2200 n \u2208 N<\/sup><\/p>\n\n\n\n

Question 11.
4n<\/sup> \u2013 3n \u2013 1 is divisible by 9<\/h3>\n\n\n\n

Answer:
Let S(n) be the statement,
4n<\/sup> \u2013 3n \u2013 1 is divisible by 9
For n = 1, 4 \u2013 3 \u2013 1 = 0 is divisible by 9
\u2234 Statement S(n) is true for n = 1.
Suppose the statement S(n) is true for n = k
Then 4k<\/sup> \u2013 3k \u2013 1 is divisible by 9.
\u2234 4k<\/sup> \u2013 3k \u2013 1 = 9t for t \u2208 N \u2026\u2026\u2026\u2026\u2026. (1)
We have to show that statement is true for n = k + 1 also.
From (1), 4k<\/sup> = 9t + 3k + 1
\u2234 4k + 1<\/sup> \u2013 3 (k + 1) \u2013 1 = 4 . 4k<\/sup> \u2013 3 (k + 1) \u2013 1
= 4 (9t + 3k + 1) \u2013 3k \u2013 3 \u2013 1
= 4 (9t) + 9k
= 9 (4t + k) divisible by 9
(\u2235 4t + k is an integer)
Hence, S(n) is true for n = k + 1 also.
\u2234 4k + 1<\/sup> \u2013 3 (k + 1) \u2013 1 is divisible by 9
\u2234 The statement is true for n = k + 1
\u2234 By the principle of Mathematical Induction.
S(n) is true for all n e K
\u2234 4n \u2013 3n \u2013 1 is divisible by 9<\/p>\n\n\n\n

Question 12.
3.52n + 1<\/sup> + 23n + 1<\/sup> is divisible by 17 (May 2012, 2008)
<\/h3>\n\n\n\n

Answer:
Let Sn be the statement
3.52n + 1<\/sup> + 23n + 1<\/sup> is divisible by 17
S(1) is 3 . 52(1) + 1<\/sup> + 23 (1)<\/sup> + 1
= 3 . 53<\/sup> + 24<\/sup> = 3 (125) + 16
= 375 + 16 = 391 is divisible by 17
Hence, S(n) is true for n = 1.
Suppose that the statement is true for n = k, then
3.52k + 1<\/sup> + 23k + 1<\/sup> is divisible by 17 and
3.52k + 1<\/sup> + 23k + 1<\/sup> = 17t for t \u2208 N
then we have to show that the result is true for n = k + 1 also
Consider 3.52(k + 1)<\/sup> + 1 + 23(k + 1)<\/sup> + 1
= 3.52k + 1 <\/sup>. 52<\/sup> + 23(k + 1)<\/sup> . 2
= (17t \u2013 23k + 1<\/sup>) 52<\/sup> + 23k + 3<\/sup> . 2
= 17t (25) \u2013 23k<\/sup> (50) + 23k<\/sup> (16)
= 17 t (25) + 23k<\/sup> (16 \u2013 50)
= 17 t (25) \u2013 34 23k<\/sup>
= 17 [25t \u2013 23k + 1<\/sup>
25t \u2013 23k + 1<\/sup> is an integer.
\u2234 3.52(k + 1)<\/sup> + 1 + 23<\/sup> (k + 1) + 1 is divisible by 17.
\u2234 The statement S(n) is true for n = k + 1 also.
\u2234 By the principle of Mathematical induction S(n) is true for \u2200n \u2208 N
\u2234 3.52n + 1<\/sup> + 23n + 1<\/sup> is divisible by 17
<\/p>\n\n\n\n

Question 13.
1.2.3 + 2.3.4 + 3.4.5 + \u2026\u2026\u2026\u2026\u2026. upto n terms = n(n+1)(n+2)(n+3)\/4. (March 2015-T.S) (Mar. 08)
<\/h3>\n\n\n\n

Answer:
The nth<\/sup> term of the given series is n (n + 1) (n + 2) and let Sn be the statement.
1.2.3 + 2.3.4 + 3.4.5 + \u2026\u2026\u2026\u2026\u2026. + n (n + 1) (n + 2)
= n(n+1)(n+2)(n+3)\/4
For n = 1
S(1) = 1.2.3 = 6
= 1(1+1)(1+2)(1+3)\/4
= 2(3)(4)\/4 = 6
\u2234 S(n) is true for n = 1
Let S(n) is true for n = k then
1.2.3 + 2.3.4 + 3.4.5 + \u2026\u2026\u2026\u2026\u2026. + k (k + 1) (k + 2)
= k(k+1)(k+2)(k+3)\/4 \u2026\u2026\u2026\u2026\u2026.. (1)
We have to prove that the result is true for n = k + 1 also.
Adding (k + 1) th term, (k + 1) (k + 2) (k + 3) both sides we get
1.2.3 + 2.3.4 + 3.4.5 + \u2026\u2026\u2026\u2026\u2026. + k(k + 1)(k + 2) + (k + 1) (k + 2) (k + 3)

\u2234 S(n) is true for n = k + 1 also.
Hence by the principle of Mathematical Induction.
S(n) is true for \u2200n \u2208 N
\u2234 1.2.3 + 2.3.4 + 3.4.5 + \u2026\u2026\u2026\u2026\u2026. + n (n + 1) (n + 2) = n(n+1)(n+2)(n+3)4<\/p>\n\n\n\n

Question 14.
13<\/sup>\/1 + 13<\/sup>+23<\/sup>\/1+3 + 13<\/sup>+23<\/sup>+33<\/sup>\/1+3+5 + \u2026\u2026\u2026\u2026\u2026\u2026 upto n terms = n\/24 (2n2 + 9n + 13). (Mar. 14, 04, 05)
<\/h3>\n\n\n\n

Answer:
The nth<\/sup> term of the given series is

Question 15.
12<\/sup> + (12<\/sup> + 22<\/sup>) + (12<\/sup> + 22<\/sup> + 32<\/sup>) + \u2026\u2026\u2026\u2026\u2026. upto n terms = n(n+1)2<\/sup>(n+2)\/12. (March 2012)
<\/p>\n\n\n\n

Answer:
Let S(n) be the statement and
nth term of series is 12<\/sup> + 22<\/sup> + 32<\/sup> + \u2026\u2026\u2026\u2026\u2026\u2026 + n2<\/sup>
\u2234 S(n) = 12<\/sup> + (12<\/sup> + 22<\/sup>) + (12<\/sup> + 22<\/sup> + 32<\/sup>) + \u2026\u2026\u2026 + (12<\/sup> + 22<\/sup> + 32<\/sup> + \u2026\u2026\u2026\u2026\u2026\u2026. + n2<\/sup>)

\u2234 S(n) is true for n = 1.
Suppose S(n) is true for n = k then
12<\/sup> + (12<\/sup> + 22<\/sup>) + (12<\/sup> + 22<\/sup> + 32<\/sup>) + \u2026\u2026\u2026 + (12<\/sup> + 22<\/sup> + 32<\/sup> + \u2026\u2026\u2026\u2026\u2026\u2026. + k2<\/sup>) = k(k+1)2(k+2)\/12
Now we have to prove that S(k + 1) is true.
So 12<\/sup> + (12<\/sup> + 22<\/sup>) + (12<\/sup> + 22<\/sup> + 32<\/sup>) + \u2026\u2026\u2026 + (12<\/sup> + 22<\/sup> + 32<\/sup> + \u2026\u2026\u2026\u2026\u2026\u2026. + k2<\/sup>) + (12<\/sup> + 22<\/sup> + 32<\/sup> + \u2026\u2026\u2026\u2026\u2026. + k2<\/sup> + (k + 1)2<\/sup>

\u2234 The statement S(n) hold for n = k + 1 also
\u2234 By the principle of Mathematical Induction S(n)is true \u2200n \u2208 N.
\u2234 12<\/sup> + (12<\/sup> + 22<\/sup>) + (12<\/sup> + 22<\/sup> + 32<\/sup>) + (12<\/sup> + 22<\/sup> + 32<\/sup> + \u2026\u2026\u2026\u2026\u2026 + n2<\/sup>) = n(n+1)2<\/sup>(n+2)\/12<\/p>\n\n\n\n

<\/p>\n","protected":false},"excerpt":{"rendered":"

Question 1.12 + 22+ 32 + \u2026\u2026\u2026\u2026. + n2 = n(n+1)(2n+1)\/6 Answer:Let S(n) be the given statement12 + 22 + 32 + \u2026\u2026\u2026\u2026. + n2 = n(n+1)(2n+1)\/6Since 12 = 1(1+1)(2+1)\/6\u21d2 1 = 1; the statement is true for n = 1Suppose the statement is true for n = k then(12 + 22 + 32 + […]<\/p>\n","protected":false},"author":2,"featured_media":2575,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[1544,15,38],"tags":[1479,1489,1484,1480,1491,1487,1481,1486,1265,1493,1488,1485,1483,1482,1490,1492,1247],"class_list":{"0":"post-2539","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-study-material-tg-inter","8":"category-telangana","9":"category-tg-inter","10":"tag-inter-1st-year-maths-1a-chapter-2","11":"tag-inter-1st-year-maths-1a-chapter-2-answers","12":"tag-inter-1st-year-maths-1a-mathematical-induction-previous-year-questions","13":"tag-mathematical-induction-problems-with-solutions","14":"tag-mathematical-induction-ts-inter-1st-year","15":"tag-mathematical-induction-ts-inter-1st-year-step-by-step-solutions","16":"tag-maths-1a-induction-method-examples","17":"tag-ts-inter-1st-year-maths-1a-chapter-2-mathematical-induction-solutions-pdf","18":"tag-ts-inter-1st-year-maths-1a-solutions","19":"tag-ts-inter-maths-1a-chapter-2-important-questions-with-answers","20":"tag-ts-inter-maths-1a-chapter-2-solutions","21":"tag-ts-inter-maths-1a-important-questions","22":"tag-ts-inter-maths-1a-pdf-download","23":"tag-ts-inter-maths-1a-textbook-solutions","24":"tag-ts-intermediate-maths-1a-chapter-2-explained-in-easy-method","25":"tag-ts-intermediate-maths-1a-mathematical-induction","26":"tag-ts-intermediate-maths-1a-study-material"},"acf":[],"yoast_head":"\r\nTS Inter 1st Year Maths 1A Solutions Chapter 2 Mathematical Induction Ex 2(a),TG Inter 1st Year Maths 1A Mathematical Induction Solutions Exercise 2(a)Using Mathematical Induction prove each of the following statement for all n \u2208 N.<\/title>\r\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\r\n<link rel=\"canonical\" href=\"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-solutions-cp-2-mathematical-induction\/\" \/>\r\n<meta property=\"og:locale\" content=\"en_US\" \/>\r\n<meta property=\"og:type\" content=\"article\" \/>\r\n<meta property=\"og:title\" content=\"TS Inter 1st Year Maths 1A Solutions Chapter 2 Mathematical Induction Ex 2(a),TG Inter 1st Year Maths 1A Mathematical Induction Solutions Exercise 2(a)Using Mathematical Induction prove each of the following statement for all n \u2208 N.\" \/>\r\n<meta property=\"og:description\" content=\"Question 1.12 + 22+ 32 + \u2026\u2026\u2026\u2026. + n2 = n(n+1)(2n+1)\/6 Answer:Let S(n) be the given statement12 + 22 + 32 + \u2026\u2026\u2026\u2026. + n2 = n(n+1)(2n+1)\/6Since 12 = 1(1+1)(2+1)\/6\u21d2 1 = 1; the statement is true for n = 1Suppose the statement is true for n = k then(12 + 22 + 32 + […]\" \/>\r\n<meta property=\"og:url\" content=\"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-solutions-cp-2-mathematical-induction\/\" \/>\r\n<meta property=\"og:site_name\" content=\"Manabadi Boards\" \/>\r\n<meta property=\"article:published_time\" content=\"2026-01-02T05:59:22+00:00\" \/>\r\n<meta property=\"article:modified_time\" content=\"2026-01-03T06:03:50+00:00\" \/>\r\n<meta property=\"og:image\" content=\"https:\/\/boardscdn.manabadi.co.in\/wp-content\/uploads\/2026\/01\/07114621\/ts-inter-1st-year-maths-1a-solutions-cp-2-mathematical-induction.jpg\" \/>\r\n\t<meta property=\"og:image:width\" content=\"900\" \/>\r\n\t<meta property=\"og:image:height\" content=\"600\" \/>\r\n\t<meta property=\"og:image:type\" content=\"image\/jpeg\" \/>\r\n<meta name=\"author\" content=\"Junaid\" \/>\r\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\r\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Junaid\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"12 minutes\" \/>\r\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-solutions-cp-2-mathematical-induction\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-solutions-cp-2-mathematical-induction\\\/\"},\"author\":{\"name\":\"Junaid\",\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/#\\\/schema\\\/person\\\/a55695e135e9dc26536b290e31b4179d\"},\"headline\":\"TS Inter 1st Year Maths 1A Solutions Chapter 2 Mathematical Induction Ex 2(a),TG Inter 1st Year Maths 1A Mathematical Induction Solutions Exercise 2(a)Using Mathematical Induction prove each of the following statement for all n \u2208 N.\",\"datePublished\":\"2026-01-02T05:59:22+00:00\",\"dateModified\":\"2026-01-03T06:03:50+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-solutions-cp-2-mathematical-induction\\\/\"},\"wordCount\":1390,\"commentCount\":0,\"image\":{\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-solutions-cp-2-mathematical-induction\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/boardscdn.manabadi.co.in\\\/wp-content\\\/uploads\\\/2026\\\/01\\\/07114621\\\/ts-inter-1st-year-maths-1a-solutions-cp-2-mathematical-induction.jpg\",\"keywords\":[\"Inter 1st Year Maths 1A Chapter 2\",\"Inter 1st Year Maths 1A Chapter 2 Answers\",\"Inter 1st Year Maths 1A Mathematical Induction Previous Year Questions\",\"Mathematical Induction Problems with Solutions\",\"Mathematical Induction TS Inter 1st Year\",\"Mathematical Induction TS Inter 1st Year Step by Step Solutions\",\"Maths 1A Induction Method Examples\",\"TS Inter 1st Year Maths 1A Chapter 2 Mathematical Induction Solutions PDF\",\"TS Inter 1st Year Maths 1A Solutions\",\"TS Inter Maths 1A Chapter 2 Important Questions with Answers\",\"TS Inter Maths 1A Chapter 2 Solutions\",\"TS Inter Maths 1A Important Questions\",\"TS Inter Maths 1A PDF Download\",\"TS Inter Maths 1A Textbook Solutions\",\"TS Intermediate Maths 1A Chapter 2 Explained in Easy Method\",\"TS Intermediate Maths 1A Mathematical Induction\",\"TS Intermediate Maths 1A Study Material\"],\"articleSection\":[\"Study Material\",\"Telangana\",\"TG Inter\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-solutions-cp-2-mathematical-induction\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-solutions-cp-2-mathematical-induction\\\/\",\"url\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-solutions-cp-2-mathematical-induction\\\/\",\"name\":\"TS Inter 1st Year Maths 1A Solutions Chapter 2 Mathematical Induction Ex 2(a),TG Inter 1st Year Maths 1A Mathematical Induction Solutions Exercise 2(a)Using Mathematical Induction prove each of the following statement for all n \u2208 N.\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-solutions-cp-2-mathematical-induction\\\/#primaryimage\"},\"image\":{\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-solutions-cp-2-mathematical-induction\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/boardscdn.manabadi.co.in\\\/wp-content\\\/uploads\\\/2026\\\/01\\\/07114621\\\/ts-inter-1st-year-maths-1a-solutions-cp-2-mathematical-induction.jpg\",\"datePublished\":\"2026-01-02T05:59:22+00:00\",\"dateModified\":\"2026-01-03T06:03:50+00:00\",\"author\":{\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/#\\\/schema\\\/person\\\/a55695e135e9dc26536b290e31b4179d\"},\"breadcrumb\":{\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-solutions-cp-2-mathematical-induction\\\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-solutions-cp-2-mathematical-induction\\\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-solutions-cp-2-mathematical-induction\\\/#primaryimage\",\"url\":\"https:\\\/\\\/boardscdn.manabadi.co.in\\\/wp-content\\\/uploads\\\/2026\\\/01\\\/07114621\\\/ts-inter-1st-year-maths-1a-solutions-cp-2-mathematical-induction.jpg\",\"contentUrl\":\"https:\\\/\\\/boardscdn.manabadi.co.in\\\/wp-content\\\/uploads\\\/2026\\\/01\\\/07114621\\\/ts-inter-1st-year-maths-1a-solutions-cp-2-mathematical-induction.jpg\",\"width\":900,\"height\":600,\"caption\":\"TS Inter 1st Year Maths 1A Solutions Chapter 2 Mathematical Induction\"},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-solutions-cp-2-mathematical-induction\\\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"TS Inter 1st Year Maths 1A Solutions Chapter 2 Mathematical Induction Ex 2(a),TG Inter 1st Year Maths 1A Mathematical Induction Solutions Exercise 2(a)Using Mathematical Induction prove each of the following statement for all n \u2208 N.\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/#website\",\"url\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/\",\"name\":\"Manabadi Boards\",\"description\":\"\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"en-US\"},{\"@type\":\"Person\",\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/#\\\/schema\\\/person\\\/a55695e135e9dc26536b290e31b4179d\",\"name\":\"Junaid\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\\\/\\\/secure.gravatar.com\\\/avatar\\\/cf93d7547c6d8b1489ad346979c0e0d181d53fe095610d9d059830de5f332e0e?s=96&d=mm&r=g\",\"url\":\"https:\\\/\\\/secure.gravatar.com\\\/avatar\\\/cf93d7547c6d8b1489ad346979c0e0d181d53fe095610d9d059830de5f332e0e?s=96&d=mm&r=g\",\"contentUrl\":\"https:\\\/\\\/secure.gravatar.com\\\/avatar\\\/cf93d7547c6d8b1489ad346979c0e0d181d53fe095610d9d059830de5f332e0e?s=96&d=mm&r=g\",\"caption\":\"Junaid\"}}]}<\/script>\r\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"TS Inter 1st Year Maths 1A Solutions Chapter 2 Mathematical Induction Ex 2(a),TG Inter 1st Year Maths 1A Mathematical Induction Solutions Exercise 2(a)Using Mathematical Induction prove each of the following statement for all n \u2208 N.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-solutions-cp-2-mathematical-induction\/","og_locale":"en_US","og_type":"article","og_title":"TS Inter 1st Year Maths 1A Solutions Chapter 2 Mathematical Induction Ex 2(a),TG Inter 1st Year Maths 1A Mathematical Induction Solutions Exercise 2(a)Using Mathematical Induction prove each of the following statement for all n \u2208 N.","og_description":"Question 1.12 + 22+ 32 + \u2026\u2026\u2026\u2026. + n2 = n(n+1)(2n+1)\/6 Answer:Let S(n) be the given statement12 + 22 + 32 + \u2026\u2026\u2026\u2026. + n2 = n(n+1)(2n+1)\/6Since 12 = 1(1+1)(2+1)\/6\u21d2 1 = 1; the statement is true for n = 1Suppose the statement is true for n = k then(12 + 22 + 32 + […]","og_url":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-solutions-cp-2-mathematical-induction\/","og_site_name":"Manabadi Boards","article_published_time":"2026-01-02T05:59:22+00:00","article_modified_time":"2026-01-03T06:03:50+00:00","og_image":[{"width":900,"height":600,"url":"https:\/\/boardscdn.manabadi.co.in\/wp-content\/uploads\/2026\/01\/07114621\/ts-inter-1st-year-maths-1a-solutions-cp-2-mathematical-induction.jpg","type":"image\/jpeg"}],"author":"Junaid","twitter_card":"summary_large_image","twitter_misc":{"Written by":"Junaid","Est. reading time":"12 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-solutions-cp-2-mathematical-induction\/#article","isPartOf":{"@id":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-solutions-cp-2-mathematical-induction\/"},"author":{"name":"Junaid","@id":"https:\/\/www.manabadi.co.in\/boards\/#\/schema\/person\/a55695e135e9dc26536b290e31b4179d"},"headline":"TS Inter 1st Year Maths 1A Solutions Chapter 2 Mathematical Induction Ex 2(a),TG Inter 1st Year Maths 1A Mathematical Induction Solutions Exercise 2(a)Using Mathematical Induction prove each of the following statement for all n \u2208 N.","datePublished":"2026-01-02T05:59:22+00:00","dateModified":"2026-01-03T06:03:50+00:00","mainEntityOfPage":{"@id":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-solutions-cp-2-mathematical-induction\/"},"wordCount":1390,"commentCount":0,"image":{"@id":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-solutions-cp-2-mathematical-induction\/#primaryimage"},"thumbnailUrl":"https:\/\/boardscdn.manabadi.co.in\/wp-content\/uploads\/2026\/01\/07114621\/ts-inter-1st-year-maths-1a-solutions-cp-2-mathematical-induction.jpg","keywords":["Inter 1st Year Maths 1A Chapter 2","Inter 1st Year Maths 1A Chapter 2 Answers","Inter 1st Year Maths 1A Mathematical Induction Previous Year Questions","Mathematical Induction Problems with Solutions","Mathematical Induction TS Inter 1st Year","Mathematical Induction TS Inter 1st Year Step by Step Solutions","Maths 1A Induction Method Examples","TS Inter 1st Year Maths 1A Chapter 2 Mathematical Induction Solutions PDF","TS Inter 1st Year Maths 1A Solutions","TS Inter Maths 1A Chapter 2 Important Questions with Answers","TS Inter Maths 1A Chapter 2 Solutions","TS Inter Maths 1A Important Questions","TS Inter Maths 1A PDF Download","TS Inter Maths 1A Textbook Solutions","TS Intermediate Maths 1A Chapter 2 Explained in Easy Method","TS Intermediate Maths 1A Mathematical Induction","TS Intermediate Maths 1A Study Material"],"articleSection":["Study Material","Telangana","TG Inter"],"inLanguage":"en-US","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-solutions-cp-2-mathematical-induction\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-solutions-cp-2-mathematical-induction\/","url":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-solutions-cp-2-mathematical-induction\/","name":"TS Inter 1st Year Maths 1A Solutions Chapter 2 Mathematical Induction Ex 2(a),TG Inter 1st Year Maths 1A Mathematical Induction Solutions Exercise 2(a)Using Mathematical Induction prove each of the following statement for all n \u2208 N.","isPartOf":{"@id":"https:\/\/www.manabadi.co.in\/boards\/#website"},"primaryImageOfPage":{"@id":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-solutions-cp-2-mathematical-induction\/#primaryimage"},"image":{"@id":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-solutions-cp-2-mathematical-induction\/#primaryimage"},"thumbnailUrl":"https:\/\/boardscdn.manabadi.co.in\/wp-content\/uploads\/2026\/01\/07114621\/ts-inter-1st-year-maths-1a-solutions-cp-2-mathematical-induction.jpg","datePublished":"2026-01-02T05:59:22+00:00","dateModified":"2026-01-03T06:03:50+00:00","author":{"@id":"https:\/\/www.manabadi.co.in\/boards\/#\/schema\/person\/a55695e135e9dc26536b290e31b4179d"},"breadcrumb":{"@id":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-solutions-cp-2-mathematical-induction\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-solutions-cp-2-mathematical-induction\/"]}]},{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-solutions-cp-2-mathematical-induction\/#primaryimage","url":"https:\/\/boardscdn.manabadi.co.in\/wp-content\/uploads\/2026\/01\/07114621\/ts-inter-1st-year-maths-1a-solutions-cp-2-mathematical-induction.jpg","contentUrl":"https:\/\/boardscdn.manabadi.co.in\/wp-content\/uploads\/2026\/01\/07114621\/ts-inter-1st-year-maths-1a-solutions-cp-2-mathematical-induction.jpg","width":900,"height":600,"caption":"TS Inter 1st Year Maths 1A Solutions Chapter 2 Mathematical Induction"},{"@type":"BreadcrumbList","@id":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-solutions-cp-2-mathematical-induction\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/www.manabadi.co.in\/boards\/"},{"@type":"ListItem","position":2,"name":"TS Inter 1st Year Maths 1A Solutions Chapter 2 Mathematical Induction Ex 2(a),TG Inter 1st Year Maths 1A Mathematical Induction Solutions Exercise 2(a)Using Mathematical Induction prove each of the following statement for all n \u2208 N."}]},{"@type":"WebSite","@id":"https:\/\/www.manabadi.co.in\/boards\/#website","url":"https:\/\/www.manabadi.co.in\/boards\/","name":"Manabadi Boards","description":"","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/www.manabadi.co.in\/boards\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"},{"@type":"Person","@id":"https:\/\/www.manabadi.co.in\/boards\/#\/schema\/person\/a55695e135e9dc26536b290e31b4179d","name":"Junaid","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/secure.gravatar.com\/avatar\/cf93d7547c6d8b1489ad346979c0e0d181d53fe095610d9d059830de5f332e0e?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/cf93d7547c6d8b1489ad346979c0e0d181d53fe095610d9d059830de5f332e0e?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/cf93d7547c6d8b1489ad346979c0e0d181d53fe095610d9d059830de5f332e0e?s=96&d=mm&r=g","caption":"Junaid"}}]}},"_links":{"self":[{"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/posts\/2539","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/comments?post=2539"}],"version-history":[{"count":4,"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/posts\/2539\/revisions"}],"predecessor-version":[{"id":2574,"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/posts\/2539\/revisions\/2574"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/media\/2575"}],"wp:attachment":[{"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/media?parent=2539"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/categories?post=2539"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/tags?post=2539"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}