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Chapter 12: 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
\( a_{14} = \frac{49}{23} \)

Step by step solution

01

- Understand the formula

The given sequence is defined by the formula: \[ a_{n} = \frac{3n+7}{2n-5} \] We need to find the specific term when \( n = 14 \).
02

- Substitute the value of 'n' in the formula

Substitute \( n = 14 \) into the formula: \[ a_{14} = \frac{3(14) + 7}{2(14) - 5} \]
03

- Simplify the expression in the numerator

Calculate the numerator: \[ 3(14) + 7 = 42 + 7 = 49 \]
04

- Simplify the expression in the denominator

Calculate the denominator: \[ 2(14) - 5 = 28 - 5 = 23 \]
05

- Divide the numerator by the denominator

Substitute the simplified expressions into \( a_{14} \) and perform the division: \[ a_{14} = \frac{49}{23} \]

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Key Concepts

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

sequence formula
In mathematics, a sequence is an ordered list of numbers that usually follows a specific pattern or rule. The sequence formula helps in identifying the terms of the sequence. For the given problem, the sequence is described by the formula: \[ a_{n} = \frac{3n+7}{2n-5} \]. It鈥檚 a function of 鈥榥鈥�, where each term in the sequence can be calculated by substituting different values of 鈥榥鈥� into the formula. Realizing how the formula works is the first step towards solving any sequence-related problem.
substitution in algebra
Substitution in algebra involves replacing a variable with a specific value. This technique is crucial when you need to find a particular term in a sequence. In this problem, we need to find the value of the term when \( n = 14 \). To do this, we substitute 14 for 鈥榥鈥� in the sequence formula: \[ a_{14} = \frac{3(14) + 7}{2(14) - 5} \]. This step is all about carefully replacing 'n' with the given value.
simplifying expressions
Simplifying algebraic expressions makes them easier to work with. When you make your substitutions, you'll often end up with expressions that can be simplified. Let鈥檚 look at our equation after substituting: \[ a_{14} = \frac{3(14) + 7}{2(14) - 5} \]. We calculate each part step by step:
  • First, the numerator: \( 3(14) + 7 \), which equals 42 + 7 = 49
  • Then, the denominator: \( 2(14) - 5 \), which equals 28 - 5 = 23
Simplifying each part individually makes the final division straightforward.
evaluating sequences
Evaluating a sequence means finding the value of a specific term after simplifying the relevant expressions. From the previous steps, we have our simplified numerator and denominator: \[ a_{14} = \frac{49}{23} \]. The last step is performing the division to get the final value. In this case, \( a_{14} \) is exactly \( \frac{49}{23} \). Making sure all arithmetic is correct ensures you get the right term from the sequence.

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