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# NCERT solutions Mathematics Class 11 chapter 4 Principle of Mathematical Induction

## Chapter 4 - Principle of Mathematical Induction

#### Pages 94 - 95

Prove the following by using the principle of mathematical induction for all n ∈ N

1 + 3 + 3^2 + ... + 3^(n – 1) =((3^n -1))/2

Q 1 | Page 94

Prove the following by using the principle of mathematical induction for all n ∈ N

1^3 +  2^3 + 3^3 + ... + n^3 = ((n(n+1))/2)^2

Q 2 | Page 94

Prove the following by using the principle of mathematical induction for all n ∈ N

1+ 1/((1+2)) + 1/((1+2+3)) +...+ 1/((1+2+3+...n)) = (2n)/(n +1)
Q 3 | Page 94

Prove the following by using the principle of mathematical induction for all n ∈ N: 1.2.3 + 2.3.4 + … + n(n + 1) (n + 2)  = (n(n+1)(n+2)(n+3))/(4(n+3))

Q 4 | Page 94

Prove the following by using the principle of mathematical induction for all n ∈ N

1.3 + 2.3^3 + 3.3^3  +...+ n.3^n = ((2n -1)3^(n+1) + 3)/4
Q 5 | Page 94

Prove the following by using the principle of mathematical induction for all n ∈ N

1.2 + 2.3 + 3.4+ ... + n(n+1) = [(n(n+1)(n+2))/3]

Q 6 | Page 94

Prove the following by using the principle of mathematical induction for all n ∈ N

1.3 + 3.5 + 5.7 + ...+(2n -1)(2n + 1) = (n(4n^2 + 6n -1))/3
Q 7 | Page 94

Prove the following by using the principle of mathematical induction for all n ∈ N: 1.2 + 2.22 + 3.22 + … + n.2n = (n – 1) 2n+1 + 2

Q 8 | Page 94

Prove the following by using the principle of mathematical induction for all n ∈ N: 1/2 + 1/4 + 1/8 + ... + 1/2^n = 1 - 1/2^n

Q 9 | Page 94

Prove the following by using the principle of mathematical induction for all n ∈ N

1/2.5 + 1/5.8 + 1/8.11 + ... + 1/((3n - 1)(3n + 2)) = n/(6n + 4)
Q 10 | Page 94

Prove the following by using the principle of mathematical induction for all n ∈ N

1/1.2.3 + 1/2.3.4 + 1/3.4.5 + ...+ 1/(n(n+1)(n+2)) = (n(n+3))/(4(n+1) (n+2))
Q 11 | Page 94

Prove the following by using the principle of mathematical induction for all n ∈ N

a + ar + ar^2 + ... + ar^(n -1) = (a(r^n - 1))/(r -1)
Q 12 | Page 95

Prove the following by using the principle of mathematical induction for all n ∈ N

(1+3/1)(1+ 5/4)(1+7/9)...(1 + ((2n + 1))/n^2) = (n + 1)^2

Q 13 | Page 95

Prove the following by using the principle of mathematical induction for all n ∈ N

(1+ 1/1)(1+ 1/2)(1+ 1/3)...(1+ 1/n) = (n + 1)

Q 14 | Page 95

Prove the following by using the principle of mathematical induction for all n ∈ N

1^2 + 3^2 + 5^2 + ... + (2n -1)^2 = (n(2n - 1) (2n + 1))/3
Q 15 | Page 95

Prove the following by using the principle of mathematical induction for all n ∈ N

1/1.4 + 1/4.7 + 1/7.10 + ... + 1/((3n - 2)(3n + 1)) = n/((3n + 1))

Q 16 | Page 95

Prove the following by using the principle of mathematical induction for all n ∈ N

1/3.5 + 1/5.7 + 1/7.9 + ...+ 1/((2n + 1)(2n +3)) = n/(3(2n +3))
Q 17 | Page 95

Prove the following by using the principle of mathematical induction for all n ∈ N: 1+2+ 3+...+n<1/8(2n +1)^2

Q 18 | Page 95

Prove the following by using the principle of mathematical induction for all n ∈ Nn (n + 1) (n + 5) is a multiple of 3.

Q 19 | Page 95

Prove the following by using the principle of mathematical induction for all n ∈ N: 102n – 1 + 1 is divisible by 11

Q 20 | Page 95

Prove the following by using the principle of mathematical induction for all n ∈ Nx2n – y2n is divisible by x y.

Q 21 | Page 95

Prove the following by using the principle of mathematical induction for all n ∈ N: 32n + 2 – 8n– 9 is divisible by 8.

Q 22 | Page 95

Prove the following by using the principle of mathematical induction for all n ∈ N: 41n – 14n is a multiple of 27.

Q 23 | Page 95

Prove the following by using the principle of mathematical induction for all n ∈ N (2+7) < (n + 3)2

Q 24 | Page 95

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