Question

1. (0 points) find the sum of the first 4 terms of the given geometric series. write your answer as a fraction or a whole number.

\\ -\frac{7}{4} - \frac{7}{12} - \frac{7}{36} - \frac{7}{108} - \frac{7}{324} - \dots \\

1. (0 points) if possible, find the sum of the infinite geometric series, and write your answer as a fraction or a whole number. if its not possible, state why.

\\ \frac{1}{5} - \frac{1}{15} - \frac{1}{45} - \frac{1}{135} - \frac{1}{405} - \dots \\

Explanation:

Step1: Identify terms for question 27

The first 4 terms are:

$$a_1 = -\frac{7}{4}, \quad a_2 = -\frac{7}{12}, \quad a_3 = -\frac{7}{36}, \quad a_4 = -\frac{7}{108}$$

Step2: Calculate the sum of 4 terms

Find a common denominator of 108:

$$S_4 = -\frac{189}{108} - \frac{63}{108} - \frac{21}{108} - \frac{7}{108} = -\frac{280}{108} = -\frac{70}{27}$$

Step3: Identify parameters for question 28

The first term and common ratio are:

$$a_1 = \frac{1}{5}, \quad r = \frac{1}{3}$$

Step4: Calculate the infinite sum

Since \(|r| < 1\), the sum is:

$$S = \frac{a_1}{1 - r} = \frac{\frac{1}{5}}{1 - \frac{1}{3}} = \frac{\frac{1}{5}}{\frac{2}{3}} = \frac{3}{10}$$

Answer:

1. \(-\frac{70}{27}\)
2. \(\frac{3}{10}\)