Question: What is the value of \( S_4 \) for this series? \[ \sum_{n=1}^{\infty} 6(2)^{n-1} \] 96 48…

What is the value of \( S_4 \) for this series?

\[ \sum_{n=1}^{\infty} 6(2)^{n-1} \]

96

48

90

84

Solution

To find \( S_4 \) for the series \[ \sum_{n=1}^{\infty} 6(2)^{n-1} \] we need to calculate the partial sum up to \( n = 4 \). The terms of the series are: For \( n = 1 \): \[ 6(2)^{1-1} = 6(2)^0 = 6(1) = 6 \] For \( n = 2 \): \[ 6(2)^{2-1} = 6(2)^1 = 6(2) = 12 \] For \( n = 3 \): \[ 6(2)^{3-1} = 6(2)^2 = 6(4) = 24 \] For \( n = 4 \): \[ 6(2)^{4-1} = 6(2)^3 = 6(8) = 48 \] Now add these terms to find \( S_4 \): \[ S_4 = 6 + 12 + 24 + 48 = 90 \] Therefore, the value of \( S_4 \) is 90.