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

1 + 3 + 3² + 3³ + …… + 3^n-1 = (3ⁿ – 1) ÷ 2

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

1³ + 2³ + 3³ + ………...+ n³ = {n (n + 1) ÷ 2}²

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

1 +  +   + …………...+  =

 

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

1.2.3 + 2.3.4 + ........+ n (n + 1) (n + 2) = {n (n + 1) (n + 2) (n + 3)} ÷ 4

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

1.3 + 2.3² + 3.3³ + n3ⁿ = {(2n - 1)3^n-1 + 3} ÷ 4

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

1.2 + 2.3 + 3.4 + …………n. (n + 1) = {n (n + 1) (n + 2)} ÷ 3

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

1.3 + 3.5 + 5.7 + …………+ (2n - 1) (2n + 1) = n (4n² + 6n - 1) ÷ 3

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

1.2 + 2.2² + 3.2² + ... + n.2ⁿ = (n – 1) 2ⁿ⁺¹ + 2

 

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

 +  +  + …………...+ (1 ÷ 2ⁿ) = 1 – (1 ÷ 2ⁿ)

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

 +  +  + …………. + [1 ÷ {(3n - 1) (3n + 1)}] = [n ÷ (6n + 4)]

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

 +  +  + …………...+ [1 ÷ {n (n + 1) (n + 2)}] = {n (n + 3)} ÷ {4(n + 1) (n + 2)}

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

a + ar + ar² + ………...+ ar^n-1 = a (rⁿ – 1) ÷ (r - 1)

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

(1 + )(1 + )(1 + )…………... {1 + {(2n + 1) ÷ n²}} = (n + 1)²

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

(1 + )(1 + )(1 + )…………. (1 + ) = (n + 1)

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

1² + 3² + 5² + ……………...+ (2n -1)² = [{n (2n - 1) (2n + 1)} ÷ 3]

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

 +  +  + …………. + [1 ÷ {(3n - 2) (3n + 1)}] =

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

 +  +  + …………. + [1 ÷ {(2n + 1) (2n + 3)}] = [n ÷ {3(2n + 3)}]

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

1 + 2 + 3 + …………+ [n < (2n + 1)² ÷ 8]

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

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

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

10^{2n – 1} + 1 is divisible by 11.

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

x²ⁿ – y²ⁿ is divisible by x + y.

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

3^{2n + 2} – 8n – 9 is divisible by 8.

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

41ⁿ – 14ⁿ is a multiple of 27.

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

(2n +7) < (n + 3)²