​ Prove that Σ_(n=1)^13(i^n+i^(n+1))= (-1 + i), n N. ​

Question

Prove that \(\displaystyle\sum^{13}_{n=1}\)(iⁿ+iⁿ⁺¹) = (-1 + i), n N.

Answer 1

L.H.S = \(\displaystyle\sum^{13}_{n=1}\)(iⁿ+iⁿ⁺¹)

= i – 1 – i + 1 + i - 1……+i -1

As, all terms will get cancel out consecutively except the first two terms. So that will get remained will be the answer.

= i - 1

L.H.S = R.H.S

Hence proved.