Find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence.${\sum }_{i=1}^8{3}^i$

Verified step by step guidance

1. Identify the type of sequence given. Since the terms are of the form \$3^i\(, this is a geometric sequence where the first term \)a = 3^1 = 3\( and the common ratio \)r = 3$.
2. Recall the formula for the sum of the first \(n\) terms of a geometric sequence: \(S_n = a \frac{r^n - 1}{r - 1}\), where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms.
3. Substitute the known values into the formula: \(a = 3\), \(r = 3\), and \(n = 8\). So, the sum is \(S_8 = 3 \frac{3^8 - 1}{3 - 1}\).
4. Simplify the denominator: \$3 - 1 = 2\(, so the sum formula becomes \)S_8 = 3 \frac{3^8 - 1}{2}$.
5. At this point, you can calculate \$3^8$, subtract 1, multiply by 3, and then divide by 2 to find the sum of the first 8 terms.

Key Concepts

Geometric Sequence

A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio. For example, in the sequence 3, 9, 27, 81, each term is multiplied by 3. Understanding this pattern is essential to identify the terms and apply the sum formula.

Sum of the First n Terms of a Geometric Sequence

The sum of the first n terms of a geometric sequence can be calculated using the formula S_n = a(1 - r^n) / (1 - r), where a is the first term and r is the common ratio. This formula simplifies adding many terms without computing each individually, which is crucial for efficiently solving the problem.

Exponents and Powers

Exponents represent repeated multiplication of a base number. In this problem, terms are expressed as powers of 3 (3^i), where i is the term index. Understanding how to work with exponents is necessary to correctly interpret the terms and apply the sum formula.