Question

Prove by method of induction, for all n ∈ N
3 + 7 + 11 + .... to n terms = n(2n + 1)

Answer 1

Answer:

Step-by-step explanation:

Given statement P(n) :

3 + 7 +........to n terms = n(2n + 1)

Put n= 1

L.H.S = 3

R.H.S = 1.(2.1 + 1) = 3

L.H.S = R.H.S

P(1) is true.

Assuming P(k) is true, where k ∈ N

P(k) : 3 + 7 +....to k terms = k(2k + 1)--------(*)

Consider P(k+1)

L.H.S = 3 + 7 + ....to k+1 term

= 3+ 7 +.......to k terms + (4k + 3)

where 4k + 3 is (k + 1) th term of given series

Using (*) in L.H.S, we get

L.H.S = k(2k + 1) + 4k + 3

= 2k² + 5k + 3

=  (k + 1)(2k + 3)

= (k + 1){2(k+1) + 1}

Hence, P(k + 1) is true .

Thus by principle of mathematical Induction P(n) is true for all n ∈ N.

Hope, it helps !

Answer 2

Step-by-step explanation:

C mini rjfjdjsajwowofkgjvj n s0