Question

Find a formula for the general term ⍺n of the sequence?

Find a formula for the general term ⍺n of the sequence, assuming that the pattern of the first few terms continues.

{-7, 14/3, -28/9, 56/27, -112/81, ...}

Answer 1

`alpha_n=-7(-2/3)^(n-1)`

Explanation:

In the given sequence, we observe that

1. Starting from first term which is negative, next term is positive and then third term is again negative and this goes on. Hence we can write it as `(-1)^n`.

2. Numertors in sequence are `{7,14,28,56,112,.......}` and as it is a geometric sequence with first term as `7` and common ratio `2`. Hence `n^(th)` term is `7xx2^(n-1)`.

3. Denominators in sequence are `{1,3,9,27,81,........}` again a geometric sequence with first term as `1` and common ratio `3`. Hence `n^(th)` term is `1xx3^(n-1)=3^(n-1)`.

Hence `n^(th)` term `alpha_n=(-1)^n(7xx2^(n-1))/3^(n-1)=-(7*(-2)^(n-1))/3^(n-1)=-7(-2/3)^(n-1)`