Problem 74 - Algebra and Trigonometry

Question

Explain how to find the general term of an arithmetic sequence.

Step-by-Step Solution

Answer

The general term of an arithmetic sequence is determined with the formula \( a_n = a_1 + (n - 1)d \), where \( a_1 \) is the first term, \( d \) is the common difference and \( n \) is the term number. Calculate the first term and the common difference and replace in the formula along with \( n \).

Step 1: Understand the Formula

Begin with the formula; \( a_n = a_1 + (n - 1)d \). Here, \( a_n \) represents the \( n^{th} \) term in the sequence, \( a_1 \) is the first term of the sequence, \( n \) is the term number and \( d \) is the common difference between each term.

Step 2: Identify the First Term

The first term (\( a_1 \)) of the sequence is usually directly given in the problem or sometimes can be determined by examining the pattern.

Step 3: Identify the Common Difference

The common difference (\( d \)) is the constant difference between successive terms of the sequence. Calculate this value by subtracting an earlier term from a later term.

Step 4: Define the Position of the Term

Recognize \( n \) as the position or term number of the sequence. For example, for finding the 5th term, \( n = 5 \). For the general term, leave \( n \) as is.

Step 5: Apply the Formula

Input identified \( a_1 \) value and \( d \) value into the formula. Keep \( n \) as is, this will give us the formula for the general term of the given sequence.