Problem 52 - Beginning and Intermediate Algebra

Question

Find the sum of the first four terms of the geometric sequence with \(a_{1}=6\) and \(r=3\)

Step-by-Step Solution

Answer

The sum of the first four terms of the given geometric sequence is \(S_4 = 240\).

Step 1: Identify the given values

We are given the first term \(a_{1}=6\) and the common ratio \(r=3\). We need to find the sum of the first four terms (\(S_4\)).

Step 2: Use the formula for the sum of the first n terms of a geometric sequence

We will use the formula \(S_n = \dfrac{a_1(r^n-1)}{r-1}\) to find the sum of the first four terms of the sequence. In our case, \(a_1 = 6\), \(r = 3\), and \(n = 4\).

Step 3: Substitute the given values into the formula

Replace the corresponding values in the formula: \(S_4 = \dfrac{6(3^4-1)}{3-1}\).

Step 4: Calculate the sum of the first four terms

Perform the calculations: \(S_4 = \dfrac{6(3^4-1)}{3-1} = \dfrac{6(81-1)}{2} = \dfrac{6(80)}{2} = \dfrac{480}{2} = 240\). Hence, the sum of the first four terms of the given geometric sequence is 240.