EXERCISE 5 1. Determine the sum of each geometric series (use an appropriate formula): (a) \( \frac{2}{27}+\frac{2}{9}+\frac{2}{3}+\ldots \). to 9 terms (b) \( -64+32-16+\ldots \). to 10 terms

For the first series \( \frac{2}{27}+\frac{2}{9}+\frac{2}{3}+\ldots \), we identify the first term \( a = \frac{2}{27} \) and the common ratio \( r = \frac{\frac{2}{9}}{\frac{2}{27}} = 3 \). The formula for the sum of the first \( n \) terms of a geometric series is \( S_n = a \frac{1-r^n}{1-r} \). Using this for 9 terms gives: \[ S_9 = \frac{2}{27} \frac{1-3^9}{1-3} = \frac{2}{27} \frac{1-19683}{-2} = \frac{2}{27} \cdot \frac{-19682}{-2} = \frac{19682}{27} \approx 728.3. \] For the second series \( -64+32-16+\ldots \), here, \( a = -64 \) and \( r = \frac{32}{-64} = -\frac{1}{2} \). Using the same formula for 10 terms: \[ S_{10} = -64 \frac{1-(-\frac{1}{2})^{10}}{1-(-\frac{1}{2})} = -64 \frac{1-\frac{1}{1024}}{1+\frac{1}{2}} = -64 \frac{1-\frac{1}{1024}}{\frac{3}{2}} = -64 \cdot \frac{2(1024-1)}{3 \cdot 1024} \approx -64 \cdot \frac{2046}{3072} \approx -42.7. \]