Questions: Given the geometric series 300+360+432+518.4+..., a. Write a geometric series formula, Sn, for the sum of the first n terms. Sn = (a1 - a1 r^n) / (1-r) b. Use the formula to find the sum of the first 8 terms, to the nearest tenth.

Given the geometric series 300+360+432+518.4+...,
a. Write a geometric series formula, Sn, for the sum of the first n terms.
Sn = (a1 - a1 r^n) / (1-r)
b. Use the formula to find the sum of the first 8 terms, to the nearest tenth.

Transcript text: Given the geometric series $300+360+432+518.4+\ldots$, a. Write a geometric series formula, $S_{n}$, for the sum of the first n terms. \[ S_{n}=\frac{a_{1}-a_{1} r^{n}}{1-r} \] b. Use the formula to find the sum of the first 8 terms, to the nearest tenth.

Solution

Solution Steps

Step 1: Identify the First Term and Common Ratio

The first term of the geometric series is given as $ a_1 = 300 $. The common ratio $ r $ can be calculated from the second term: \[ r = \frac{360}{300} = 1.2 \]

Step 2: Use the Geometric Series Sum Formula

We apply the geometric series sum formula for the first $ n $ terms: \[ S_n = \frac{a_1(1 - r^n)}{1 - r} \] Substituting $ a_1 = 300 $, $ r = 1.2 $, and $ n = 8 $: \[ S_8 = \frac{300(1 - (1.2)^8)}{1 - 1.2} \]

Step 3: Calculate the Sum

After performing the calculations, we find: \[ S_8 \approx 4949.7254 \] Rounding this to the nearest tenth gives: \[ S_8 \approx 4949.7 \]