MHB How Do You Solve a Geometric Sum with Alternating Signs?

[Kola Citron]

Hey!

I'm stuck again and not sure how to solve this question been at it for a few hours. Any help is appreciated as always.

Q: (1) Let the sum S = 3- 3/2 + 3/4 - 3/8 + 3/16 - 3/32 +...- 3/128. Determine integers a , n and a rational number k so that...(Image)



(2 )And then calculate S using the geometric sum formula.

Thank you!

 

[skeeter]

common ratio is $r = -\dfrac{1}{2}$

note $128 = 2^7$

first term is $a = 3$

$\displaystyle S = 3 \sum_{j=0}^7 \left(-\dfrac{1}{2}\right)^j$

you can calculate the sum ...

 

[HOI]

Given that it is a "geometric sum", a+ ar+ ar^2+ ...,, you can determine r, the "common ratio" by just dividing the second term by the first: ar/a= r. In this problem that is (-3/2)/3= -1/2.