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Chapter 13: Problem 23

Find the indicated term for each sequence. $$ a_{n}=(n+1)(2 n+3) ; \quad a_{8} $$

Short Answer

Expert verified
The 8th term of the sequence is 171.

Step by step solution

01

Identify the general formula

The general formula for the sequence is given as \(a_{n}=(n+1)(2 n+3)\).
02

Substitute the value of n

To find the 8th term in the sequence, substitute \(n=8\) into the general formula: \(a_{8}=(8+1)(2(8)+3)\).
03

Simplify the expression inside the parentheses

Calculate the values inside the parentheses: \(8+1=9\) and \(2(8)+3=16+3=19\).
04

Multiply the simplified values

Multiply the results obtained: \(9 \times 19 = 171\).
05

Write the final term

The 8th term of the sequence \(a_{n}\) is \(a_{8}=171\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

General Formula
In arithmetic sequences, we often have a general formula that helps to calculate any term in the sequence. For instance, the given sequence has the formula:
\(a_{n} = (n+1)(2n+3)\). This represents every term in the sequence where \(n\) is the position of the term we want to find.
The general formula simplifies the task of finding terms at different positions without needing to list out all the terms one by one.
Sequence Term Calculation
Calculating a specific term in a sequence involves substituting the position number into the general formula. If you want to find the 8th term in the given sequence, you substitute \(n = 8\) into the formula.

So, we substitute to get: \[a_{8} = (8+1)(2(8)+3)\].
This process makes identifying any term straightforward and saves time.
Substitution and Simplification
After substituting the value of \(n\), the next step is simplifying the expression. For our example of finding the 8th term:
First, solve inside the parentheses: \(8+1=9\) and \(2(8)+3=19\).
This gives us: \[a_{8} = 9 \times 19\].
Then, multiply these simplified values. Finally, \[9 \times 19 = 171\].
Simplification helps to break down complex steps into simpler, more manageable calculations.
Term Identification
Identifying the specific term once the calculations are done is the last step. From our previous simplifications, we determined: \[a_{8} = 171\].
This means that the 8th term in the sequence is 171.
Understanding how to identify terms in an arithmetic sequence allows for quick and accurate answers to mathematical problems.

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Most popular questions from this chapter

Find the indicated term of each binomial expansion. The term with \(x^{8} y^{2}\) in \(\left(2 x^{2}+3 y\right)^{6}\)

Write the first four terms of each binomial expansion. $$ (2 p-3 q)^{11} $$

Solve each problem. The number of mutual funds operating in the United States each year during the period 2012 through 2016 is given in the table. To the nearest whole number, what was the average number of mutual funds operating per year during the given period? $$ \begin{array}{|c|c|} \hline \text { Year } & \text { Number of Mutual Funds } \\ 2012 & 8744 \\ 2013 & 8972 \\ 2014 & 9258 \\ 2015 & 9517 \\ 2016 & 9511 \\ \hline \end{array} $$

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