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Chapter 15: Problem 45

For a particular sequence, suppose you are asked to find \(S_{15}\). What are you finding?

Short Answer

Expert verified
When you are asked to find \(S_{15}\) for a particular sequence, you are finding the sum of the first 15 terms of that sequence. To do this, you need to identify the general term of the sequence, determine the appropriate summation formula depending on the type of sequence (arithmetic or geometric), and substitute the given values to calculate the sum.

Step by step solution

01

Identify the Sequence

First, we need to identify the given sequence and its general term, usually represented as \(a_n\). It will be provided in the problem or derived using the given information.
02

Determine the Summation Formula

Next, we need a formula to sum the first 15 terms of the sequence, generally represented as: \[S_n = a_1 + a_2 + a_3 + ... + a_n\] Where \(S_n\) represents the sum of the first n terms and \(a_i\) represents the ith term of the sequence. Depending on the type of sequence, different formulas may be used. For arithmetic sequences, the formula is: \[S_n = \frac{n(a_1 + a_n)}{2}\] For geometric sequences, the formula is: \[S_n = a_1\frac{1 - r^n}{1 - r}\] Where r is the common ratio.
03

Calculate \(S_{15}\) using the Summation Formula

Once the appropriate summation formula is identified, we will substitute the given values for \(n\), \(a_1\), \(a_n\), and/or \(r\), depending on the formula. In this case, we want to find \(S_{15}\), meaning we need to find the sum of the first 15 terms, so we plug \(n = 15\) into the formula: For arithmetic sequences: \[S_{15} = \frac{15(a_1 + a_{15})}{2}\] For geometric sequences: \[S_{15} = a_1\frac{1 - r^{15}}{1 - r}\]
04

Calculate the Value of \(S_{15}\) Following the Sequence's Properties

Solve for \(S_{15}\) using the specific values for \(a_1\), \(a_{15}\), and/or \(r\) provided in the problem. The result will be the sum of the first 15 terms of the sequence. Once the above steps are followed for the given sequence, the value of \(S_{15}\) will be found, which represents the sum of the first 15 terms of the sequence.

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Most popular questions from this chapter

Evaluate each binomial coefficient. $$\left(\begin{array}{l}8 \\\5\end{array}\right)$$

Use the binomial theorem to expand each expression. $$(a-3)^{4}$$

A home purchased for \(\$ 140,000\) increases in value by \(5 \%\) per year. a) Find the general term of the geometric sequence that models the future value of the house. b) How much is the home worth 8 yr after it is purchased? (Hint: Think carefully about what number to substitute for \(n .\) ) Round the answer to the nearest dollar.

Use the formula for \(S_{n}\) to find the sum of the terms of each geometric sequence. $$\sum_{i=1}^{5} 2\left(\frac{1}{3}\right)^{i}$$

Find the sum of the terms of the infinite geometric sequence, if possible. $$-12,8,-\frac{16}{3}, \frac{32}{9}, \dots$$
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