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Chapter 14: Problem 15

Find the indicated term for each sequence. $$ a_{n}=\frac{3 n+7}{2 n-5} ; \quad a_{14} $$

Short Answer

Expert verified
The term \( a_{14} \) is \( \frac{49}{23} \).

Step by step solution

01

Identify the general term equation

The general term for the sequence is given by the formula \[ a_n = \frac{3n + 7}{2n - 5} \].
02

Determine the value of n

For this problem, you need to find the value of the term when \( n = 14 \).
03

Substitute n with 14 in the equation

Substitute \( n = 14 \) into the general term formula:\[ a_{14} = \frac{3(14) + 7}{2(14) - 5} \]
04

Simplify the numerator

Calculate the value of the numerator:\[ 3(14) + 7 = 42 + 7 = 49 \]
05

Simplify the denominator

Calculate the value of the denominator:\[ 2(14) - 5 = 28 - 5 = 23 \]
06

Divide the simplified numerator by the denominator

Now, divide the simplified numerator by the simplified denominator to find the term:\[ a_{14} = \frac{49}{23} \]

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Key Concepts

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

General Term
To solve sequence problems, understanding the general term is crucial. The general term is a formula that allows you to find any term in the sequence. For our example, the general term is given by:

\[ a_n = \frac{3n + 7}{2n - 5} \]

This formula represents how the terms in the sequence are formed. The variable n represents the position of the term in the sequence. By plugging different values of n into the equation, you can find the value of any term.
Substitution
Once you know the general term, the next step is substitution. This means replacing the variable n with the specific term number you are looking to find. In this case, we need to find the 14th term of the sequence, so we substitute n with 14:

\[ a_{14} = \frac{3(14) + 7}{2(14) - 5} \]

This step is all about inputting the right number into the general term formula. It transforms the general formula into something you can directly calculate.
Simplification
The last step involves simplification. It means performing arithmetic operations to make the expression simpler. Begin with the numerator and denominator separately:

  • Numerator: Calculate \( 3(14) + 7 = 42 + 7 = 49 \)

  • Denominator: Calculate \( 2(14) - 5 = 28 - 5 = 23 \)


Finally, divide the simplified numerator by the simplified denominator:

\[ a_{14} = \frac{49}{23} \]

Now, you have the value of the 14th term in the sequence. This step-by-step simplification ensures that complex problems become easier to manage.

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Most popular questions from this chapter

Write out the first five terms of each sequence. $$ a_{n}=n+4 $$

If the given sequence is geometric, find the common ratio \(r\). If the sequence is not geometric, say so. See Example 1 . $$ \frac{1}{3}, \frac{2}{3}, \frac{3}{3}, \frac{4}{3}, \ldots $$

Evaluate the indicated term for each arithmetic sequence. $$ 2,4,6, \ldots ; \quad a_{24} $$

If the given sequence is geometric, find the common ratio \(r\). If the sequence is not geometric, say so. See Example 1 . $$ \frac{5}{7}, \frac{8}{7}, \frac{11}{7}, 2, \dots $$

Use the formula for \(a_{n}\) to find the general term of each arithmetic sequence. $$ -10,-5,0, \ldots $$
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