ncert solutions for class 9,10,11 and 12 : Prove the following by using the principle of mathematical induction for all n ∈ N

NCERT Solution For Class 11 Maths

Chapter -4 Mathematical Induction

The Principle of Mathematical Induction

Suppose there is a given statement P(n) involving the natural number n such that
(i) The statement is true for n = 1, i.e., P(1) is true, and
(ii) If the statement is true for n = k (where k is some positive integer), then the statement is also true for n = k + 1, i.e., truth of P(k) implies the truth of P (k + 1).
Then, P(n) is true for all natural numbers n.

Exercise solutions

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

1 + 3 + 3² + ... + 3^{n – 1} = (3ⁿ-1)/2
Answer

Prove the following by using the principle of mathematical induction for all n ∈ N
1³ + 2³ + 3³ + … +n³= {(n(n+1)/2}²

Answer

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)
Answer

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

Answer

Prove the following by using the principle of mathematical induction for all n ∈ N
1.3 + 2.3² + 3.3³ +…+ n.3ⁿ={(2n − 1)3 ⁿ^{+ 1} + 3}/4

Answer

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

Answer

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(4n²+6n -1)/3

Answer

Prove the following by using the principle of mathematical induction for all n ∈ N
1.2 + 2.2²+ 3.2² + ...+n.2ⁿ = (n-1) 2^{n + 1} + 2.

Answer

Prove the following by using the principle of mathematical induction for all n ∈ N
1/2 +1/4 + 1/8 + … + 1/2ⁿ = 1 – 1/2ⁿ

^Answer

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)

Answer
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)

Answer
a + ar + ar² +…+ ar^n-1 = a (rⁿ − 1)/(r-1)

Answer
(1 + 3/1)(1 + 5/4)(1 + 7/9)…(1 + {(2n+1)/n²} = (n+1)²

^Answer
(1 + 1/1)(1 + 1/2)(1 + 1/3)…(1 + 1/n) = (n+1)

Answer
1²+ 3² + 5² + …+ (2n-1)² = n (2 n − 1) (2 n + 1)/3

Answer
1/1.4 + 1/4.7+1/7.10 +… + 1/(3 n − 2)(3 n + 1) = n/(3 n + 1)

Answer
1/3.5 + 1/5.7 + 1/7.9 + … + 1/(2 n + 1)(2 n + 3) = n/{3(2 n + 3)}

Answer
1 + 2 + 3 +…+ n < 1/8(2n + 1)².

Answer
n (n + 1) (n + 5) is a multiple of 3.

Answer
10²ⁿ^{- 1} + 1 is divisible by 11.

Answer
x²ⁿ - y²ⁿ is divisible by x + y.

Answer
3²ⁿ⁺² - 8n - 9 is divisible by 8.

Answer
41ⁿ - 14ⁿ is a multiple of 27.

Answer
(2n + 7) < (n + 3)²

^Answer