Use the formula for the sum of the first n terms of a geometric sequence to find the indicated sum. *sigma notation* i=1 start, 6 end, 5^i formula

Verified step by step guidance

1. Identify 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.
2. Recognize that the given sequence is \( 5^i \) from \( i = 1 \) to \( i = 6 \). Here, the first term \( a = 5^1 = 5 \), the common ratio \( r = 5 \), and the number of terms \( n = 6 \).
3. Substitute the values into the formula: \( S_6 = 5 \frac{5^6 - 1}{5 - 1} \).
4. Calculate \( 5^6 \) and subtract 1 from it to find the numerator.
5. Divide the result by 4 (since \( r - 1 = 5 - 1 = 4 \)) to find the sum of the sequence.

Key Concepts

Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. For example, in the sequence 2, 6, 18, 54, the common ratio is 3. Understanding geometric sequences is essential for applying the formula for the sum of their terms.

Sum of the First n Terms

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 S_n is the sum, a is the first term, r is the common ratio, and n is the number of terms. This formula allows for efficient calculation of the total when dealing with a finite number of terms in a geometric series.

Sigma Notation

Sigma notation is a concise way to represent the sum of a sequence of terms. It uses the Greek letter sigma (Σ) to indicate summation, specifying the index of summation, the starting and ending values, and the expression to be summed. In the given question, sigma notation is used to denote the sum of the terms generated by the formula 5^i from i=1 to i=6.