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Chapter 14: Problem 16

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

Short Answer

Expert verified
\(a_{16} = \frac{71}{56}\).

Step by step solution

01

Understand the sequence formula

The given sequence formula is \(a_{n} = \frac{5n-9}{3n+8}\). We need to find the term \(a_{16}\).
02

Substitute the value of n

Substitute \(n = 16\) into the formula: \(a_{16} = \frac{5(16) - 9}{3(16) + 8}\).
03

Simplify the numerator

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

Simplify the denominator

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

Combine the results

Substitute the simplified parts back into the fraction: \(a_{16} = \frac{71}{56}\).
06

Simplify the fraction if possible

Check if \(\frac{71}{56}\) can be simplified. Since 71 and 56 have no common factors other than 1, \(\frac{71}{56}\) 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.

algebraic sequences
An algebraic sequence is a list of numbers where each term is generated by a specific algebraic formula. In our case, the formula given is \(a_{n} = \frac{5n-9}{3n+8}\). Understanding this formula helps us predict the value of any term in the sequence. Sequences can often be found in a wide range of mathematical and real-life problems. By using the formula, we can efficiently find the nth term without manually listing all preceding terms.

Let's break it down:
  • **Numerator**: The term \(5n - 9\) tells us how rapidly the numerator grows as n increases.
  • **Denominator**: Similarly, \(3n + 8\) tells us the growth rate of the denominator.
Knowing the structure of the formula provides insight into the behavior of the sequence.
term evaluation
Evaluating the indicated term of a sequence means substituting the given value of n into the sequence formula. For instance, to find \(a_{16}\), you simply plug 16 into \(a_{n} = \frac{5n-9}{3n+8}\). This process is straightforward if done step-by-step:

1. Substitute n = 16: \(a_{16} = \frac{5(16) - 9}{3(16) + 8}\)

2. Simplify the numerator:
  • Calculate \(5(16) - 9\): \( 80 - 9 = 71\)

3. Simplify the denominator:
  • Calculate \(3(16) + 8\): \(48 + 8 = 56\)

4. Combine results: This gives us \(a_{16} = \frac{71}{56}\).
simplifying fractions
Simplifying the fractions in sequence formulas makes them easier to understand. It involves reducing a fraction to its simplest form by ensuring the numerator and denominator have no common factors other than 1. For example, for \(a_{16} = \frac{71}{56}\), we check to see if 71 and 56 share any factors. Since 71 is a prime number and 56 = 2 * 2 * 2 * 7, they share no common divisors. Thus, \(\frac{71}{56}\) is already simplified.

Key steps to simplify any fraction:
  • 1. Find the greatest common divisor (GCD) of the numerator and denominator.
  • 2. Divide both numerator and denominator by the GCD.
  • 3. If the GCD is 1, the fraction is already in its simplest form.
Simplifying helps in better comparison and understanding of terms in sequences.

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Most popular questions from this chapter

Write out the first five terms of each sequence. $$ a_{n}=2^{n} $$

If the given sequence is geometric, find the common ratio \(r\). If the sequence is not geometric, say so. See Example 1 . $$ 1,-\frac{1}{2}, \frac{1}{4},-\frac{1}{8}, \ldots $$

Find the indicated term for each sequence. $$ a_{n}=3 n-7 ; \quad a_{12} $$

Solve each problem involving an ordinary annuity. Derrick Ruffin deposits \(\$ 10,000\) at the end of each year for 12 yr in an account paying \(5 \%\) compounded annually. He then puts the total amount on deposit in another account paying \(6 \%\) compounded semiannually for another 9 yr. Find the final amount on deposit after the entire 21 -yr period.

Find a general term \(a_{n}\) for the given terms of each sequence. $$ 7,14,21,28, \dots $$
See all solutions

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