Find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence.${\sum }_{i=1}^6{\left(\frac{1}{2}\right)}^{i+1}$

Verified step by step guidance

1. Identify the geometric sequence and the number of terms. Here, the sequence is given by the general term \(a_i = \left(\frac{1}{2}\right)^{i+1}\), and the number of terms is \(n = 6\).
2. Rewrite the general term to express it in the form \(a r^{i-1}\), where \(a\) is the first term and \(r\) is the common ratio. For \(i=1\), the first term is \(a = \left(\frac{1}{2}\right)^{1+1} = \left(\frac{1}{2}\right)^2\).
3. Determine the common ratio \(r\) by comparing consecutive terms. Since the exponent increases by 1 each time, \(r = \frac{1}{2}\).
4. Use the formula for the sum of the first \(n\) terms of a geometric sequence: \(S_n = a \frac{1 - r^n}{1 - r}\)
5. Substitute the values of \(a\), \(r\), and \(n\) into the formula: \(S_6 = \left(\frac{1}{2}\right)^2 \frac{1 - \left(\frac{1}{2}\right)^6}{1 - \frac{1}{2}}\) This expression represents the sum you need to calculate.

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 ratio. For example, in the sequence 1, 1/2, 1/4, 1/8, ..., each term is multiplied by 1/2. Understanding the structure of geometric sequences is essential to identify the terms and apply the sum formula correctly.

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 the process of adding multiple terms without computing each individually, which is crucial for efficiently solving the given sum.

Index Manipulation and Term Identification

Correctly interpreting the index in the summation, especially when the exponent involves (i + 1), is important to identify the first term and the common ratio. Adjusting the index helps to rewrite the sum in a standard geometric form, ensuring the sum formula is applied accurately.