In Exercises 31–36, find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence. 6 Σ (i = 1) (1/2)^(i + 1)

Verified step by step guidance

1. Identify the sequence as a geometric sequence with the general term \( a_i = \left(\frac{1}{2}\right)^{i+1} \).
2. Determine the first term \( a_1 \) by substituting \( i = 1 \) into the general term: \( a_1 = \left(\frac{1}{2}\right)^{1+1} = \left(\frac{1}{2}\right)^2 \).
3. Identify the common ratio \( r \) of the geometric sequence, which is \( \frac{1}{2} \).
4. Use the formula for the sum of the first \( n \) terms of a geometric sequence: \( S_n = a_1 \frac{1-r^n}{1-r} \), where \( n = 6 \).
5. Substitute \( a_1 = \left(\frac{1}{2}\right)^2 \), \( r = \frac{1}{2} \), and \( n = 6 \) into the sum formula to find \( S_6 \).

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. In this case, the common ratio is 1/2, which indicates that each term is half of the previous term. Understanding this concept is crucial for identifying the pattern in the sequence and calculating the sum.

Sum of a Geometric Series

The sum of the first n terms of a geometric series can be calculated using the formula S_n = a(1 - r^n) / (1 - r), where '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 sum without needing to add each term individually, which is essential for solving the given problem.

Index Notation

Index notation is a way of representing the terms in a sequence using a variable, typically 'i', which indicates the position of each term. In the given sum, the index 'i' starts at 1 and goes to 6, affecting the exponent in the term (1/2)^(i + 1). Understanding how to interpret and manipulate index notation is vital for correctly applying the sum formula.