Question

How do you write the first five terms of the geometric sequence `a_1=6, a_(k+1)=-3/2a_k` and determine the common ratio and write the nth term of the sequence as a function of n?

Answer 1

See details below

Explanation:

A geometric sequence is defined as: an ordered sequence of number such that each term is calculated multipliying the prior term by a constant number called ratio. In mathematical terms

`a_(k+1)=r·a_k` or if you want `a_k=r·a_(k-1)`

In our case `a_(k+1)=-3/2·a_k`. So, the ratio is `-3/2`.

Lets calculate:

`a_1=6`
`a_2=-3/2·6=-9`
`a_3=-3/2·(-9)=+27/2`
`a_4=-3/2·27/2=-81/4`
`a_5=-3/2·(-81/4)=+243/8`

The general term is given by `a_n=a_1·r^(n-1)`...in our case

`a_n=6·(-3/2)^(n-1)`

Each term with a odd position will be positive and each term in a even postion will be negative (oscillating)