stuck on geometric series questions when common ration is negative?

Question

Hi all, this probably sounds mad, but when I am working out the sum of n terms in a geometric series, when the common ratio is negative I am getting the answers wrong every time and cannot see where i am going wrong.

The book I have has proven the formula

Sn = [a(1-r^n)]/[1-r]

where Sn = sum of n terms
a = first term and r = common ratio

Now take the question when the first term = 5 and common ratio = -2 find the sum of 10 terms

s I put in

{S10} = 5(1 - (-2^10)) / 1 - - 2
= 5(1 - (-1024) / 3
= 5(1025) / 3
= 5125/3

Now the answer is wrong, because it should be -5115/3 which would then give the correct answer to me. The only place I can see how they get this is to have 5(1 - 1024) but I cannot see how this can be....please help its driving me mad!!!!

Answer 1

S = a1 (1 - r^n) / (1 - r)

S = 5 (1 - (-2)^10) / (1 - (-2) )

S = 5 (1 - 1024) / (1 + 2)

S = 5 (-1023) / 3

S = -5115/3


the mistake you made is missing parrenthese

it's supposed to be: 1 - (-2)^10 (not 1 - (-2^10)

-2^10 is -1 (2)^10 = -1024

(-2)^10 = 1024

Answer 2

I don't see any problem, the formula is right and is good for every r <>1. In your example,

S_10 = 5 ( 1 - (-2)^10)/(1 -(-2)) = 5 (1 - 1024)/(3) = 5 *(-1023)/3 = -1705.

And , indeed, -5115/3 = -1705

Your mistake is here
= 5(1 - (-1024) / 3

(-2)^10 = 1024 and not -1024

Answer 3

{S10} = 5(1 - (-2^10)) / 1 - - 2
This first step is where you've gone wrong
it should be
=5(1 - ((-2))^10)) / 1 - - 2
in other words you need to raise (-2) to the power 10
and not have 2 to the power 10 with a minus sign
(-2)^10 is 1024 which is what you need
to get the answer you want (that is the correct one!)

Answer 4

You are leaving the minus sign out of the ratiio when raising it to a power.
(-2)^10 = + 1024

Answer 5

(-2)^10 = +1024 not -1024 as you have used. Make that change and you have it.