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Chapter 12: Problem 17

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

Short Answer

Expert verified
The 8th term, \( a_{8} \), is 171.

Step by step solution

01

- Identify the general term in the sequence

The general term of the sequence is given by the formula \[ a_{n}=(n+1)(2n+3) \]
02

- Substitute the value of n

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

- Simplify the expression

First, calculate the values inside the parentheses: \[ 8+1 = 9 \] and \[ 2(8)+3 = 16+3 = 19 \]. Then multiply the results together: \[ a_{8} = 9 \times 19 \]
04

- Compute the final value

Multiply 9 by 19: \[ 9 \times 19 = 171 \]. So, \( a_{8} = 171 \).

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Key Concepts

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

General Term of a Sequence
The general term of a sequence provides a formula that generates the terms of the sequence. In other words, it tells you how to find any term in the sequence if you know its position (n).
For example, in our exercise, the general term is given as:
  • \[a_n = (n+1)(2n+3)\]

Here, n represents the position of the term in the sequence. This formula allows us to find the nth term by plugging in the value of n.
The importance of understanding the general term is that it provides a consistent and predictable pattern for calculating sequence terms.
Substitution Method
The substitution method involves replacing variables with their actual values to compute specific terms or solve equations.
To find the 8th term in our sequence, we replace n with 8 in the general term formula:
  • \[a_8 = (8+1)(2 \times 8 + 3)\]

This step translates the formula into a specific term calculation by using a particular value.
Substitution is critical as it simplifies complex expressions, making numerical calculations direct and efficient.
Simplifying Expressions
Simplifying expressions means reducing them to their simplest form. It involves performing all possible basic arithmetic operations to make an expression easier to understand or solve.
In our exercise, we simplified the substituted expression:
  • \[a_8 = (8+1)(2 \times 8 + 3)\]

First, calculate inside the parentheses:
  • \[8 + 1 = 9\]
  • \[2 \times 8 + 3 = 16 + 3 = 19\]

Combining these, you get:
  • \[a_8 = 9 \times 19\]

Simplifying expressions helps you get to the final answer in a clear and organized way.
Multiplication of Integers
Multiplication of integers is a basic arithmetic operation that combines two numbers to get a product. It follows the commutative property, meaning the order of numbers doesn鈥檛 affect the result.
In our case, after substitution and simplification, we multiplied the integers:
  • \[a_8 = 9 \times 19\]

Performing the multiplication, we get:
  • \[9 \times 19 = 171\]

This fundamental operation is essential in various aspects of mathematics and is pivotal in finding terms in sequences efficiently.

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Most popular questions from this chapter

Solve each problem. The total assets of mutual funds operating in the United States, in billions of dollars, for each year during the period 2004 through 2008 are shown in the table. What were the average assets per year during this period? \(\begin{array}{|c|c|}\hline \text { Year } & {\text { Assets (in billions of dollars) }} \\ {2004} & {8107} \\ {2005} & {8905} \\ {2006} & {10,397} \\\ {2007} & {12,000} \\ {2008} & {9601} \\ \hline\end{array}\)

Find the arithmetic mean for each collection of numbers. $$ 2,1,4,8,3,7,10,8,0 $$

Solve each problem involving an ordinary annuity. A father opened a savings account for his daughter on her first birthday, depositing \(1000. Each year on her birthday he deposits another \)1000, making the last deposit on her 21st birthday. If the account pays 4.4% interest compounded annually, how much is in the account at the end of the day on the daughter鈥檚 21st birthday?

Evaluate each expression. $$ 6 ! $$

Solve each problem. The number of mutual funds operating in the United States available to investors each year during the period 2004 through 2008 is given in the table. To the nearest whole number, what was the average number of funds available per year during the given period? \(\begin{array}{|c|c|}\hline & {\text { Number of }} \\ \hline \text { Year } & {\text { Funds Available }} \\ {2004} & {8041} \\ {2005} & {7975} \\\ {2006} & {8117} \\ {2007} & {8024} \\ {2008} & {8022} \\ \hline\end{array}\)
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