Finding the Term of an Arithmetic Sequence: A Comprehensive Guide

When dealing with arithmetic sequences, it is often necessary to identify the term at a specific position given the initial term and the common difference. This article explores the step-by-step process to determine the 21st term of an arithmetic sequence that starts with 2 and has a common difference of 3, and provides detailed explanations for each approach.

Understanding the Problem

The problem at hand involves an arithmetic sequence where the first term a1 is 2, and the common difference d is 3. The goal is to determine which term in the sequence is equal to 62. To solve this, we will use the general formula for the nth term of an arithmetic sequence:

Step 1 - Using the General Formula

Let's start by using the general formula for the nth term:

an a1 (n-1)d

In this sequence:

a1 2 d 3 an 62

Substituting the given values into the formula:

62 2 (n-1)3

Step 2 - Simplifying the Equation

To find n, let's simplify the equation step by step:

62 2 3(n-1) 62 2 3n - 3 62 3n - 1 62 1 3n 63 3n 21 n

Thus, the 21st term of the arithmetic sequence is 62.

Alternative Approaches

There are several alternative methods to find the term in an arithmetic sequence. Let's explore another approach:

Method 2 - Using the Difference Between Terms

We can also solve this problem using the common difference approach. Since the common difference is 3, we can express the term 62 in terms of the initial term 2 and the common difference:

62 2 3(n-1)

Step 1 - Rewrite the Equation

Let's rewrite the equation and solve for n:

62 2 3(n-1) 62 2 3n - 3 62 3n - 1 Step 2 - Simplify the Equation

Now, let's simplify and solve for n:

62 1 3n 63 3n 21 n

Similar to the previous approach, the 21st term of the arithmetic sequence is 62.

Generalizing the Solution

From the process above, we can derive a general formula for finding the term in an arithmetic sequence:

Tn a1 (n-1)d

For our specific sequence:

Tn 2 (n-1)3 Tn 3n - 1

This general term rule,

Tn 3n - 1

can be used to find any term in the sequence.

Conclusion

Understanding arithmetic sequences and their terms is crucial for solving various mathematical problems. The process involves identifying the common difference, substituting into the general formula, and simplifying. Whether using the general formula or the difference between terms, the method remains consistent. The key takeaway is to practice these techniques to ensure proficiency in solving problems related to arithmetic sequences.