Find the sum of the following geometric series.

Ask by Henry Phillips. in the United States

Jan 26,2025

Question

Find the sum of the following geometric series. $$ 3+2.55+2.1675+1.842375+\ldots $$

UpStudy AI · Accepted Answer

To find the sum of the infinite geometric series $$ 3 + 2.55 + 2.1675 + 1.842375 + \ldots $$, follow these steps:

1. **Identify the first term and the common ratio:**
   - **First term ($$ a $$)**: $$ 3 $$
   - **Common ratio ($$ r $$)**: To find $$ r $$, divide the second term by the first term:
     $$
     r = \frac{2.55}{3} = 0.85
     $$
     To confirm, check the ratio of subsequent terms:
     $$
     \frac{2.1675}{2.55} = 0.85 \quad \text{and} \quad \frac{1.842375}{2.1675} = 0.85
     $$
     This confirms that the common ratio is $$ 0.85 $$.

2. **Use the formula for the sum of an infinite geometric series:**
   $$
   S = \frac{a}{1 - r}
   $$
   Plugging in the values:
   $$
   S = \frac{3}{1 - 0.85} = \frac{3}{0.15} = 20
   $$

**Final Answer:**  
The sum of the series is **20**.
The sum of the series is 20.

Find the sum of the following geometric series.

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