Question

Find the sum of the geometric sequence. (1 point) 1, 1/2, 1/4, 1/8, 1/16

Answer 1

Sn  = a1 * (1 - r^n)
                 ----------
                   1 - r

n = 5   so answer  =

   1 * (1 -  (1/2)^5
        ---------------  =       31/16  
               1 - 1/2
           

Answer 2

That finite sequence we can just add up, using 16 as a common denominator

$1 + 1/2 + 1/4 + 1/8+1/16 = 16/16 + 8/16 + 4/16+2/16+1/16 = (16+8+4+2+1)/16=31/16$

They probably want you to use the formula

$S_n = \sum_{k=0}^{n-1} r^k = \dfrac{1 - r^n}{1-r}$

That's the special case of a geometric series where the first term is one.  If it's not we multiply the answer by the first term.

In our example, r=1/2 and n=5, meaning five terms.

$S_5 = \dfrac{ 1- (1/2)^5}{1- (1/2)} = \dfrac{1-1/32}{1/2}=\dfrac{31/32}{1/2}=\dfrac{31}{15}$

Same answer, math works!

If the question was asking about the infinite geometric series, that's just a little bigger:

$S = \dfrac{1}{1-r} = \dfrac{1}{1 - \frac 1 2} = 2$