Sum of Getometric Sequence with alternating signs

[notSomebody]

Homework Statement

5^2 - 5^3 + 5^4 - ... + (-1)^k*5^k whre k is an integer with k >= 2

Homework Equations

The Attempt at a Solution

I know (5^(k-1) - 5^2)/2 gives you the sum if they were all positive. I tried multiplying it by (-1)^k or something but that just changes the sign. I wish I could give you more but I can't.

 

[HallsofIvy]

A geometric sequence is $\sum_{n=0}^N ar^n$. And the sum is:
$$\frac{1- r^{N+1}}{1- r}$$
r does not have to be positive. Your sequence has a= 1, r= -5.

 

[notSomebody]

$$\frac{1- (-5)^{2+1}}{1- (-5)}$$ = 21 though and not 25

 

[SammyS]

The sum that HallsofIvy gave includes (-5)⁰ and (-5)¹

 

[HallsofIvy]

There are two ways to handle the fact that your sum starts with $r^2$ rather than $r^0= 1$.

1) Factor out an $r^2$ $(-5)^2+ (-5)^3\cdot\cdot\cdot+ (-5)^k= (-5)^2(1+ (-5)+ \cdot\cdot\cdot+ (-5)^{k-2})$

Use the formula I gave with n= k- 2 and then multiply by $(-5)^2= 25$.

2) Use the formula with n= k and then subtract of $(-5)^0+ (-5)^1= 1- 6= -4$.