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Chapter 11: Problem 15

Find the indicated term of each sequence. $$ a_{n}=(3 n+1)(2 n-5) ; a_{9} $$

Short Answer

Expert verified
The indicated term \( a_{9} \) is 364.

Step by step solution

01

Understand the Sequence Formula

The given sequence is defined by the formula: \[ a_{n}=(3n+1)(2n-5) \] which represents the general term of the sequence.
02

Substitute the Given Term Number

To find the indicated term, substitute \( n = 9 \) into the formula:\[ a_{9} = (3(9)+1)(2(9)-5) \]
03

Simplify Inside the Parentheses

First, simplify the expressions inside the parentheses:\[ 3(9) + 1 = 27 + 1 = 28 \]\[ 2(9) - 5 = 18 - 5 = 13 \]
04

Multiply the Simplified Expressions

Multiply the results to find the value of the indicated term:\[ a_{9} = 28 \times 13 = 364 \]

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Key Concepts

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

Sequence Formula
A sequence formula helps to find any term in a sequence without listing all terms. In this problem, the sequence formula is given as: \[ a_{n} = (3n + 1)(2n - 5) \]. This formula represents a general term in the sequence. By substituting the value of n, we can find the specific term we need.

This sequence formula combines two linear expressions, which then get multiplied together to provide the term value. The variables within the parentheses are expanded according to multiplication rules.
Substitution
Substitution is used to find the value of a specific term in a sequence. To start, identify the term you want to find (e.g., the 9th term). Here's how you use substitution in our problem:
  • Identify the term number: Here, it is given as n = 9.
  • Substitute n = 9 into the sequence formula: \[ a_{9} = (3(9) + 1)(2(9) - 5) \].

Substitution replaces the variable with the given term number, setting the stage for the next steps鈥攕implification and multiplication.
Simplification
Simplification is the process of reducing expressions into simpler forms. After substitution, you need to simplify inside the parentheses first:
  • Calculate \[ 3(9) + 1 = 27 + 1 = 28 \].
  • Calculate \[ 2(9) - 5 = 18 - 5 = 13 \].

This step will leave you with simpler numbers to work with for the final calculation.
Multiplication
The last step is to multiply the simplified results. Multiply the numbers obtained from the previous simplification step to get the value of the specific term:
  • Multiply 28 and 13: \[ a_{9} = 28 \times 13 \].

Using basic multiplication:
  • 28 multiplied by 13 equals 364.

The 9th term in the sequence is therefore 364.

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Most popular questions from this chapter

Monthly Earnings. Suppose that you accepted a job for the month of February \((28\) days) under the following conditions. You will be paid \(\$ 0.01\) the first day, \(\$ 0.02\) the second, \(\$ 0.04\) the third, and so on, doubling your previous day's salary each day. How much would you earn?

Determine whether each infinite geometric series has a limit.If a limit exists, find it. $$0.37+0.0037+0.000037+\cdots$$

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