Given a geometric sequence where the first term is 5 and the common ratio is 3, find the sum of the first 4 terms.

To find the sum of the first four terms of a geometric sequence with a first term of \( 5 \) and a common ratio of \( 3 \), follow these steps: ### Geometric Sequence Basics A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant called the **common ratio** (\( r \)). The formula for the \( n \)-th term of a geometric sequence is: \[ a_n = a_1 \times r^{(n-1)} \] where: - \( a_n \) is the \( n \)-th term, - \( a_1 \) is the first term, - \( r \) is the common ratio, - \( n \) is the term number. ### Sum of the First \( n \) Terms The sum of the first \( n \) terms (\( S_n \)) of a geometric sequence can be calculated using the formula: \[ S_n = a_1 \times \frac{r^n - 1}{r - 1} \] provided that \( r \neq 1 \). ### Applying the Formula Given: - First term (\( a_1 \)) = 5 - Common ratio (\( r \)) = 3 - Number of terms (\( n \)) = 4 Plugging these values into the sum formula: \[ S_4 = 5 \times \frac{3^4 - 1}{3 - 1} \] Calculate \( 3^4 \): \[ 3^4 = 81 \] Subtract 1: \[ 81 - 1 = 80 \] Divide by \( 3 - 1 = 2 \): \[ \frac{80}{2} = 40 \] Multiply by the first term: \[ S_4 = 5 \times 40 = 200 \] ### Verification by Listing the Terms For clarity, let's list the first four terms and sum them: 1. **First term (\( a_1 \))**: \[ a_1 = 5 \] 2. **Second term (\( a_2 \))**: \[ a_2 = a_1 \times r = 5 \times 3 = 15 \] 3. **Third term (\( a_3 \))**: \[ a_3 = a_2 \times r = 15 \times 3 = 45 \] 4. **Fourth term (\( a_4 \))**: \[ a_4 = a_3 \times r = 45 \times 3 = 135 \] Now, sum these terms: \[ S_4 = a_1 + a_2 + a_3 + a_4 = 5 + 15 + 45 + 135 = 200 \] ### Conclusion The sum of the first four terms of the geometric sequence is **200**.