Question

write an equation for the nth term of the arithmetic sequence.

1. 4, 13, 22, 31, ...

Explanation:

Step1: Find the common difference \( d \)

The common difference \( d \) in an arithmetic sequence is the difference between consecutive terms. For the sequence \( 4, 13, 22, 31, \dots \), we calculate \( d = 13 - 4 = 9 \), \( 22 - 13 = 9 \), \( 31 - 22 = 9 \). So, \( d = 9 \).

Step2: Recall the formula for the \( n \)-th term of an arithmetic sequence

The formula for the \( n \)-th term \( a_n \) of an arithmetic sequence is \( a_n = a_1 + (n - 1)d \), where \( a_1 \) is the first term and \( d \) is the common difference.

Step3: Identify \( a_1 \) and substitute \( a_1 \) and \( d \) into the formula

Here, the first term \( a_1 = 4 \) and \( d = 9 \). Substituting these values into the formula, we get:

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Answer:

The equation for the \( n \)-th term of the arithmetic sequence \( 4, 13, 22, 31, \dots \) is \( a_n = 9n - 5 \)