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Chapter 11: Problem 13

Find the indicated term of each sequence. $$ a_{n}=5 n+3 ; a_{8} $$

Short Answer

Expert verified
The 8th term of the sequence is 43.

Step by step solution

01

Understand the formula

The given formula for the sequence is \(a_{n} = 5n + 3\). This formula tells us that the general term of the sequence depends on the value of \(n\).
02

Identify the term to find

The problem asks us to find \(a_{8}\), which means \(n = 8\).
03

Substitute the value into the formula

Substitute \(n = 8\) into the formula \(a_{n} = 5n + 3\). This gives us \(a_{8} = 5(8) + 3\).
04

Calculate the result

Perform the multiplication and addition: \(a_{8} = 5 \times 8 + 3 = 40 + 3 = 43\).

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Key Concepts

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

sequence formula
The sequence formula is a mathematical expression used to find any term within a sequence. In our exercise, the sequence formula is given as:\[a_{n} = 5n + 3\]. This specific formula defines an arithmetic sequence, a type of sequence where the difference between consecutive terms is constant. To use a sequence formula effectively, always identify the variables within the formula. In this case, 'n' represents the term's position in the sequence.
substitution
Substitution is the process of replacing a variable in an expression with a specific value. In our exercise, we need to find the 8th term of the sequence, denoted as \(a_{8}\). This means we substitute \(n = 8\) into our sequence formula \(a_{n} = 5n + 3\). When substituting, replace the variable 'n' with the given value, ensuring you maintain the structure of the formula.Thus, our new equation becomes:\[a_{8} = 5(8) + 3\]
arithmetic operations
Arithmetic operations refer to basic mathematical operations such as addition, subtraction, multiplication, and division. These operations are essential when calculating sequence terms. After substitution, we have:\[a_{8} = 5(8) + 3\] To solve, you need to perform the multiplication first, as per the order of operations: \[5 \times 8 = 40\] Then, add the remaining number: \[40 + 3 = 43\] Hence, \(a_{8} = 43\)
general term
The general term of a sequence is the formula that allows you to find any term in the sequence. For any arithmetic sequence, the general term formula reliably reveals each term based on its position.We already know that: \[a_{n} = 5n + 3\] represents our general term. By substituting any value for 'n', we can find the term that corresponds to that position in the sequence. In our specific example, substituting \(n = 8\) leads us to calculate the 8th term, giving us \(a_{8} = 43\). This flexibility makes the general term formula a powerful tool in sequence analysis.

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