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Chapter 14: Problem 7

Find the indicated term of each sequence. $$ a_{n}=2 n-3 ; a_{8} $$

Short Answer

Expert verified
The 8th term of the sequence is 13.

Step by step solution

01

Identify the General Formula

The general term of the sequence is given by the equation: a_n = 2n - 3.
02

Substitute the Given Value of n

To find the 8th term in the sequence, substitute the value of 8 for n in the general formula: a_8 = 2(8) - 3.
03

Simplify the Expression

Perform the multiplication and subtraction in the expression: a_8 = 16 - 3.
04

Find the Result

After performing the operations, we get: a_8 = 13.

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Key Concepts

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

general term
When we talk about sequences, the 'general term' is a formula that allows us to find any term in the sequence. This formula is usually written as \(a_n\), where \(n\) represents the position of the term in the sequence. For example, in the sequence given by \(a_{n} = 2n - 3\), \(a_1\) would be the first term, \(a_2\) the second term, and so on. Each term depends on the position number (\(n\)) and is calculated using the general formula.
sequence formula
The sequence formula is a specific type of mathematical rule that defines how each term in a sequence is calculated. It's like a recipe that tells you exactly how to generate each subsequent term. In our example, the sequence formula is \(a_{n} = 2n - 3\). This simple linear formula shows us that for any position \(n\) in the sequence, you multiply the position number by 2 and then subtract 3 to get the term value. These formulas can vary in complexity, but understanding how to work with them is essential for solving sequence problems.
substitution in sequences
Substitution is a method used to calculate a specific term in a sequence by replacing the variable in the formula with an actual number. In our example, we need to find the 8th term of the sequence. Starting with the formula \(a_{n} = 2n - 3\), we substitute \(n\) with 8, because we are looking for \(a_8\). So it becomes \(a_8 = 2(8) - 3\). By substituting 8 for \(n\), we've transformed the general formula into a specific calculation for the 8th term in the sequence.
simplifying algebraic expressions
Once you've substituted the value into the sequence formula, you often need to simplify the expression to find the answer. For our problem, after substituting \(n\) with 8, we have the expression \(a_8 = 2(8) - 3\). Simplifying this involves multiplying and then subtracting. First, compute the multiplication: \(2 \times 8 = 16\). Next, perform the subtraction: \(16 - 3\). The result is \(a_8 = 13\). Simplifying algebraic expressions involves basic arithmetic operations to arrive at the final value, helping us see the clear result of our calculations.

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