Use the formula for the sum of the first n terms of a geometric sequence to find the indicated sum.
$\sum_{i = 1}^{6} 5^{i}$

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Identify the geometric sequence and its parameters. Here, the terms are given by $ 5^i $ for $ i = 1 $ to $ 6 $. The first term $ a $ is $ 5^1 = 5 $, and the common ratio $ r $ is 5 because each term is multiplied by 5 to get the next term.

Recall the formula for the sum of the first $ n $ terms of a geometric sequence: \[ S_n = a \times \frac{r^n - 1}{r - 1} \] where $ a $ is the first term, $ r $ is the common ratio, and $ n $ is the number of terms.

Substitute the known values into the formula: $ a = 5 $, $ r = 5 $, and $ n = 6 $. This gives \[ S_6 = 5 \times \frac{5^6 - 1}{5 - 1} \].

Simplify the denominator $ 5 - 1 = 4 $ to rewrite the sum as \[ S_6 = 5 \times \frac{5^6 - 1}{4} \].

At this point, you can calculate $ 5^6 $, subtract 1, then multiply by 5 and divide by 4 to find the sum of the series.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 5, 25, 125, ..., each term is multiplied by 5. Understanding this helps identify the pattern and apply the correct formulas.

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 efficiently computes the total without adding each term individually.

Sigma Notation

Sigma notation (∑) is a concise way to represent the sum of a sequence of terms. It specifies the index of summation, the starting and ending values, and the general term formula. Understanding sigma notation helps translate the problem into a sum that can be evaluated using formulas.

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