Problem 116 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the indicated partial sum of the series. \(\sum_{n=1}^{\infty} 8\left(-\frac{1}{4}\right)^{n}\) Fourth partial sum

Step-by-Step Solution

Answer

After doing the calculations, the fourth partial sum of the series is \(\frac{224}{21}\).

Step 1: Identify the terms in the series

Here, a geometric series is given. The first term of the series \(a = 8*(-\frac{1}{4})\) and the common ratio \(r = -\frac{1}{4}\)

Step 2: Use the formula for the sum of a geometric series

The formula for the sum of a geometric series is \(S = \frac{a(1-r^n)}{1-r}\) where \(S\) is the sum of the first \(n\) terms, \(a\) is the first term, \(r\) is the common ratio and \(n\) is the number of terms.

Step 3: Calculate the fourth partial sum using the formula

Substitute \(a = 8*(-\frac{1}{4})\), \(r = -\frac{1}{4}\), and \(n = 4\) into the formula: \(S = \frac{a(1-r^n)}{1-r} = \frac{8*(-\frac{1}{4})*(1-(-\frac{1}{4})^4)}{1-(-\frac{1}{4})}\)