Question

Find the 85 th term of the sequence \(5,12,19,26, \ldots\)

Short answer

The 85th term is 593

Step-by-step solution

Identify the type of sequence

Recognize that the given sequence is arithmetic because the difference between consecutive terms is constant.

Determine the first term and common difference

The first term \(a_1\) of the sequence is 5. To find the common difference \(d\), subtract the first term from the second term: \(d = 12 - 5 = 7\).

Use the formula for the nth term of an arithmetic sequence

The formula for the nth term \(a_n\) of an arithmetic sequence is: \[ a_n = a_1 + (n-1)d \]

Plug in the values

Substitute \(a_1 = 5\), \(d = 7\), and \(n = 85\) into the formula: \[ a_{85} = 5 + (85-1) \times 7 \]

Perform the calculations

Simplify the expression: \[ a_{85} = 5 + 84 \times 7 = 5 + 588 = 593 \]

Key concepts

nth term formula

Understanding the formula for the nth term of an arithmetic sequence can make finding specific terms much easier. An arithmetic sequence is a list of numbers where the difference between consecutive terms is always the same. The formula for the nth term, denoted as \(a_n\), is given by: \[ a_n = a_1 + (n-1)d \] In this formula:
\(a_n\) is the term you want to find.
\(a_1\) is the first term of the sequence.
\(d\) is the common difference between the terms.
\(n\) represents the position of the term in the sequence.
By plugging in the known values, you can easily find any term in the sequence. For instance, if you know the first term and the common difference, you can plug in those values along with the position of the desired term to find it quickly.

common difference

The common difference is a crucial part of an arithmetic sequence. It is the amount by which each term increases or decreases to get to the next term. To find the common difference, simply subtract the first term from the second term in the sequence.
For instance, in the sequence \(5, 12, 19, 26, \ldots\), we subtract: \[ d = 12 - 5 = 7 \] This tells us that each term increases by 7.
Knowing the common difference helps us understand the pattern in the sequence and allows us to use the nth term formula effectively.

sequence

A sequence is an ordered list of numbers. In an arithmetic sequence, this list follows a specific pattern where each term has a constant difference from the previous term.
For example, in the sequence \(5, 12, 19, 26, \ldots\), the pattern is clear because each term increases by 7.
Sequences are fundamental in mathematics, helping us understand patterns and predict future terms. By recognizing the type of sequence and its rules, such as an arithmetic sequence with a common difference, you can solve various problems related to growth, progression, and prediction easily.
Remember, identifying the type of sequence is the first step in tackling any sequence problem!