Let `S_n = 1.5 +2.5^2 +3.5^3 + ... + n.5^n =1/16[(4n - 1)5^a +b]`, then

By method of induction prove that 1.3 + 2.5 + 3.7 +...+ n (2n + 1) = n/6 (n + 1) (4n + 5) for all n in N

Prove the following by the method of induction for all n in N : 5 + 5^2 + 5^3 + .. + 5^n = 5/4 (5^n - 1)

1.3 + 2.4 + 3.5 + …… + n(n+2) =( n(n+1)(2n+7))/6

if A= {5^(n)-4n-1: n in N} and B={16(n-1):n in N} then

Find S_n , if S_n = 1.3 + 2.4 + 3.5 + … upto n terms.

Evaluate: lim_ (n rarr oo) (1 ^ (4) + 2 ^ (4) + 3 ^ (4) + ... + n ^ (4)) / (n ^ (5)) - lim_ (n rarr oo) (1 ^ (3) + 2 ^ (3) + ... + n ^ (3)) / (n ^ (5))

Evaluate: lim_ (n rarr oo) (1 ^ (4) + 2 ^ (4) + 3 ^ (4) + ... + n ^ (4)) / (n ^ (5)) - lim_ (n rarr oo) (1 ^ (3) + 2 ^ (3) + ... + n ^ (3)) / (n ^ (5))

lim_ (n rarr oo) (2.3 ^ (n + 1) -3.5 ^ (n + 1)) / (2.3 ^ (n) + 3.5 ^ (n)) =

Find S_n for the series. 1.2.3.4 + 2.3.4.5 + 3.4.5.6 +…

Prove the following by the method of induction for all n in N : 1/1.3 + 1/3.5 + 1/5.7+...+ 1 / ((2n-1)(2n+1)) = n / (2n+1)