Question

Prove each of the following for all n ≥ 1 by the Principle of Mathematical Induction.

(a). 12 + 32 + 52 + ... + (2n − 1)2 =

n(2n−1)(2n+1)

3

(b). 1.3 + 2.4 + 3.5 + ... + n(n + 2) = n(n+1)(2n+7)

6

(c). ∑n

i=1

1

i(i+1) =

n

n+1

(d). ∑n

i=1 2

i−1 = 2n − 1

(e). ∑n

i=1 i(2i

) = 2 + (n − 1)2n+1

(f). ∑n

n=1(i)(i!) = (n + 1)! − 1

(g). 13 + 23 + 23 + ... + n

3 =

n

2

(n+1)2

4​

Answer 1

Answer:

you do on your own ok bye