Question

PLEASE HELP!!! Use sigma notation to represent the following series for the first 12 terms.

Answer 1

If you notice, in the sum the terms are 2, -8, 32, -128....

we can always get the "common ratio" of a geometric sequence by dividing the "following term by the previous term", namely like in this case say -8/2, which is -4, so r = -4, and we know the first term is 2.

so, notice, the pattern will then be

 $\bf \stackrel{2(-4)^0}{2}~~\stackrel{2(-4)^1}{-8}~~\stackrel{2(-4)^2}{32}~~\stackrel{2(-4)^3}{-128}\qquad \implies \qquad \sum\limits_{k=0}^{11}~2(-4)^k$