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Chapter 14: Problem 22

Find the indicated term for each sequence. $$ a_{n}=\frac{5 n-9}{3 n+8} ; \quad a_{16} $$

Short Answer

Expert verified
The 16th term of the sequence is \( \frac{71}{56} \).

Step by step solution

01

Identify the general term

The sequence is given by the general term \(a_n = \frac{5n - 9}{3n + 8}\). This formula represents the nth term of the sequence.
02

Substitute the value of n

We are asked to find the 16th term of the sequence. To do this, substitute \(n = 16\) into the general term formula: \(a_{16} = \frac{5(16) - 9}{3(16) + 8}\).
03

Simplify the numerator

Calculate the numerator by substituting \(n = 16\): \(5(16) - 9 = 80 - 9 = 71\).
04

Simplify the denominator

Calculate the denominator by substituting \(n = 16\): \(3(16) + 8 = 48 + 8 = 56\).
05

Compute the fraction

Now, we have the fraction \(a_{16} = \frac{71}{56}\). This fraction is already in its simplest form.

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Key Concepts

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

general term
In the given sequence, the general term is represented by the formula: \(a_n = \frac{5n - 9}{3n + 8}\). This formula is crucial because it allows us to calculate any term in the sequence by simply substituting the position number (n) into the formula. By doing this, we can find the value of any specific term without having to list out the entire sequence. Notice that the general term is a fraction where both the numerator and the denominator are linear functions of n. Understanding this concept is key when working with sequences.
substitution
Substitution is the process of replacing a variable with a specific value. In our exercise, we are asked to find the 16th term of the sequence. To do this, we substitute the value of n with 16 in the general term. This means wherever we see n in our formula \(a_n = \frac{5n - 9}{3n + 8}\), we replace it with 16: \(a_{16} = \frac{5(16) - 9}{3(16) + 8}\). Substitution helps us transform an abstract formula into a specific value or answer.
simplification of fractions
Simplification of fractions involves reducing the fraction to its lowest terms. In our exercise, after substituting n with 16, the formula becomes \(a_{16} = \frac{71}{56}\). To simplify, we check if the numerator and denominator have any common factors. If they do, we divide both by the greatest common factor. However, in this case, \( \frac{71}{56} \), the fraction is already in its simplest form, meaning 71 and 56 have no common factors other than 1. Therefore, no further simplification is needed.

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Most popular questions from this chapter

Write the first five terms of each sequence. $$ a_{n}=6(-1)^{n+1} $$

Fill in each blank with the correct response. The number of terms in the geometric sequence \(1,2,4, \ldots, 2048\) is ______.

If the given sequence is arithmetic, find the common difference \(d .\) If the sequence is not arithmetic, say so. See Example 1. \(10,5,0,-5,-10, \ldots\)

Write each series as a sum of terms and then find the sum. $$ \sum_{i=1}^{6}(-1)^{i} \cdot 2 $$

Write each series using summation notation. $$ 7+8+9+10+11 $$
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