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Ch. 8 - Sequences, Induction, and Probability
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 8 - Sequences, Induction, and ProbabilityProblem 35
Chapter 9, Problem 35

Find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence.∑i=16(12)i+1\(\sum\)_{i=1}^{6} \(\left\)(\(\frac{1}{2}\]\right\))^{i+1}

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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 \(n = 6\).
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 \(a = \left(\frac{1}{2}\right)^{1+1} = \left(\frac{1}{2}\right)^2\).
Determine the common ratio \(r\) by comparing consecutive terms. Since the exponent increases by 1 each time, \(r = \frac{1}{2}\).
Use the formula for the sum of the first \(n\) terms of a geometric sequence: \(S_n = a \frac{1 - r^n}{1 - r}\)
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.

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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 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.
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Geometric Sequences - Recursive 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 the process of adding multiple terms without computing each individually, which is crucial for efficiently solving the given sum.
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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.
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Example 1
Related Practice
Textbook Question

Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of the sequence. Find a7 when a1 = 2, r = 3

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Textbook Question

Find each indicated sum. ∑i=14(−12)i\(\sum\)_{i=1}^{4} \(\left\)(-\(\frac{1}{2}\]\right\))^{i}

Textbook Question

Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of the sequence. Find a(sub 6) when a(sub 1) = 16, r = 1/2

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Textbook Question

Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of the sequence. Find a5 when a1 = -3, r = 2

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Write the first three terms in each binomial expansion, expressing the result in simplified form. (x2 + 1)16