Question

How do you write the first five terms of the geometric sequence `a_1=5, a_(k+1)=-2a_k` and determine the common ratio and write the nth term of the sequence as a function of n?

Answer 1

first 5 terms

`{5,-10,20,-40,80}`

common ratio

`r=-2`

nth term

`a_n=5(-2)^(n-1)`

Explanation:

`a_1=5`

`a_(k+1)=-2a_k`

`a_1=5`

`a_2=-2a_1=-2xx5=-10`

`a_3=-2a_2=-2xx-10=20`

`a_4--2a_3=-2xx20=-40`

`a_5=-2a_4=-2xx-40=80`

first 5 terms

`{5,-10,20,-40,80}`

common ratio

is `a_(k+1)/a_k=(-2a_k)/a_k`

`r=-2`

nth term of any Gp

`a_n=a_1r^(n-1)`

`a_n=5(-2)^(n-1)`

Answer 2

`a_1 = 5` and `a_(k + 1) = -2a_k`

so, `a_2 = -2a_1`

=> `a_2 = -10`

Also, `a_3 = -2a_2`

i.e. `a_3 = 20`

so, our terms are `5, -10, 20, -40,...`

Hence, 1st term 5 and common ratio -2

Now, nth term is `ar^(n - 1)`

i.e. `5(-2)^(n - 1))`

or, `(-5/2)(-2^n)`

With n = 7 we have:

`(-5/2)(-2^7) = 320`

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