Question

Evaluate the geometric series described below.a1 = -1, r = 4, n = 8

Answer 1

First let's see the expression for a geometric series:

$\sum ^n_{k\mathop=0}a\cdot r^k$

In this problem r=4 and n=8. a is not given, however, you do have the second term of the series:

$a_1=-1$

This means that, when k=1, the term equals 1:

$a\cdot4^1=-1$

So, we can figure out the value of a from here:

$a=-\frac{1}{4}$

So, the complete geometric series given is:

$\sum ^8_{k\mathop=0}\frac{-1}{4}\cdot4^k=\sum ^8_{k\mathop=0}-4^{k-1}$

And finally you just have to evaluate the sum:

$\sum ^8_{k\mathop=0}-4^{k-1}=-(4^{-1}+4^0+4^1+4^2+4^3+4^4+4^5+4^6+4^7)=-21845.25^{}^{}^{}^{}^{}$