1) Evaluate the geometric series described: \(-5, \ 15, \ -45, \ 135, \ -405, \ ..., \ n \ = \ 7\)
\(\color{red}{r \ = \ \frac{15}{-5} \ = \ -3}\)
\(\color{red}{S_n \ = \ \frac{a(r^n \ - \ 1)}{r \ - \ 1} \ = \ \frac{-5((-3)^7 \ - \ 1)}{-3 \ - \ 1} \ = \ \frac{-5(-2187 \ - \ 1)}{-3 \ - \ 1} \ = \ 2735}\)
2) Evaluate the geometric series described: \(9, \ 18, \ , \ 36, \ 72, \ ..., \ n \ = \ 9\)
\(\color{red}{r \ = \ \frac{18}{9} \ = \ 2}\)
\(\color{red}{S_n \ = \ \frac{a(r^n \ - \ 1)}{r \ - \ 1} \ = \ \frac{9(2^9 \ - \ 1)}{2 \ - \ 1} \ = \ 9(512 \ - \ 1) \ = \ 4599}\)
3) Evaluate the geometric series described: \(\sum_{n \ = \ 1}^{n \ = \ 9} \ 7(-2)^{n \ - \ 1}\)
\(\color{red}{\sum_{n \ = \ 1}^{n \ = \ 9} \ 7(-2)^{n \ - \ 1} \ ⇒ \ r \ = \ -2, \ a \ = \ 7}\)
\(\color{red}{S_n \ = \ \frac{a(r^n \ - \ 1)}{r \ - \ 1} \ = \ \frac{7((-2)^9 \ - \ 1)}{-2 \ - \ 1} \ = \ \frac{7(-512 \ - \ 1)}{-3} \ = \ 1197}\)
4) Evaluate the geometric series described: \(\sum_{n \ = \ 1}^{n \ = \ 5} \ 5(3)^{n \ - \ 1}\)
\(\color{red}{\sum_{n \ = \ 1}^{n \ = \ 5} \ 5(3)^{n \ - \ 1} \ ⇒ \ r \ = \ 3, \ a \ = \ 5}\)
\(\color{red}{S_n \ = \ \frac{a(r^n \ - \ 1)}{r \ - \ 1} \ = \ \frac{5(3^5 \ - \ 1)}{3 \ - \ 1} \ = \ \frac{5(243 \ - \ 1)}{2} \ = \ 605}\)
5) Evaluate the geometric series described: \(\sum_{n \ = \ 1}^{n \ = \ 6} \ 3(-\frac{1}{2})^{n \ - \ 1}\)
\(\color{red}{\sum_{n \ = \ 1}^{n \ = \ 6} \ 3(-\frac{1}{2})^{n \ - \ 1} \ ⇒ \ r \ = \ -\frac{1}{2} \ , \ a \ = \ 3}\)
\(\color{red}{S_n \ = \ \frac{a(r^n \ - \ 1)}{r \ - \ 1} \ = \ \frac{3((-\frac{1}{2})^5 \ - \ 1)}{-\frac{1}{2} \ - \ 1} \ = \ \frac{3(-\frac{1}{32} \ - \ 1)}{-\frac{3}{2}} \ = \ \frac{66}{32} \ = \ \frac{33}{16}}\)
6) Evaluate the geometric series described: \(\sum_{n \ = \ 1}^{n \ = \ 4} \ 8(\frac{1}{3})^{n \ - \ 1}\)
\(\color{red}{\sum_{n \ = \ 1}^{n \ = \ 4} \ 8(\frac{1}{3})^{n \ - \ 1} \ ⇒ \ r \ = \ \frac{1}{3} \ , \ a \ = \ 8}\)
\(\color{red}{S_n \ = \ \frac{a(r^n \ - \ 1)}{r \ - \ 1} \ = \ \frac{8((\frac{1}{3})^4 \ - \ 1)}{\frac{1}{3} \ - \ 1} \ = \ \frac{8(\frac{1}{81} \ - \ 1)}{-\frac{2}{3}} \ = \ \frac{320}{27}}\)
7) Evaluate the geometric series described: \(1, \ 4, \ 16, \ 64, \ 256, \ ..., \ n \ = \ 6\)
\(\color{red}{r \ = \ \frac{4}{1} \ = \ 4}\)
\(\color{red}{S_n \ = \ \frac{a(r^n \ - \ 1)}{r \ - \ 1} \ = \ \frac{1((4)^6 \ - \ 1)}{4 \ - \ 1} \ = \ \frac{4096 \ - \ 1}{3} \ = \ 1365}\)
8) Evaluate the geometric series described: \(1024, \ 256, \ 64, \ 16, \ 4, \ ..., \ n \ = \ 6\)
\(\color{red}{r \ = \ \frac{256}{1024} \ = \ \frac{1}{4}}\)
\(\color{red}{S_n \ = \ \frac{a(r^n \ - \ 1)}{r \ - \ 1} \ = \ \frac{1024((\frac{1}{4} \ )^6 \ - \ 1)}{\frac{1}{4} \ - \ 1} \ = \ \frac{1024(\frac{1}{4096} \ - \ 1}{-\frac{3}{4}} \ = \ 1365}\)
9) Evaluate the geometric series described: \(7, \ 14, \ 28, \ 56, \ 112, \ ..., \ n \ = \ 4\)
\(\color{red}{r \ = \ \frac{14}{7} \ = \ 2}\)
\(\color{red}{S_n \ = \ \frac{a(r^n \ - \ 1)}{r \ - \ 1} \ = \ \frac{7((2)^4 \ - \ 1)}{2 \ - \ 1} \ = \ 7(16 \ - \ 1) \ = \ 105}\)
10) Evaluate the geometric series described: \(54, \ 18, \ 6, \ 2, \ \frac{2}{3}, \ ..., \ n \ = \ 4\)
\(\color{red}{r \ = \ \frac{18}{54} \ = \ \frac{1}{3}}\)
\(\color{red}{S_n \ = \ \frac{a(r^n \ - \ 1)}{r \ - \ 1} \ = \ \frac{54((\frac{1}{3})^4 \ - \ 1)}{\frac{1}{3} \ - \ 1} \ = \ 80}\)
