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How to Find Terms of an Arithmetic Progression?

Published in Arithmetic Progressions 3 mins read

To find terms of an arithmetic progression (AP), you can use a specific formula based on the first term and the common difference.

Understanding Arithmetic Progressions

An arithmetic progression is a sequence of numbers where the difference between any two consecutive terms is constant. This constant difference is called the "common difference."

Formula for the nth Term

The most common way to find a term in an arithmetic progression is by using the following formula:

an = a + (n - 1)d

Where:

  • an is the nth term of the AP (also sometimes called the general term or last term if the sequence is finite)
  • a is the first term of the AP
  • n is the position of the term you want to find (e.g., 1 for the first term, 2 for the second term, etc.)
  • d is the common difference between consecutive terms

How to Use the Formula

Here is a step-by-step approach to use the formula:

  1. Identify the First Term (a): Determine the starting number of the sequence.
  2. Find the Common Difference (d): Subtract any term from the term that follows it in the sequence. For example, if the sequence is 2, 5, 8, 11..., then d = 5-2 = 3.
  3. Determine the Term Position (n): Decide which term you want to find (e.g., the 5th term, the 10th term, etc.).
  4. Plug the Values into the Formula: Substitute the values of a, n, and d into the formula an = a + (n - 1)d.
  5. Solve for an: Calculate the result to get the nth term.

Example

Let's find the 7th term of the arithmetic progression: 3, 7, 11, 15,...

  1. First Term (a): a = 3
  2. Common Difference (d): d = 7 - 3 = 4
  3. Term Position (n): n = 7
  4. Apply Formula: a₇ = 3 + (7 - 1) * 4
  5. Solve: a₇ = 3 + (6) * 4 = 3 + 24 = 27

Therefore, the 7th term of this AP is 27.

Practical Insights

  • This formula can be used to find any term in an arithmetic progression without needing to calculate all the preceding terms.
  • If the common difference d is positive, the terms of the AP increase.
  • If the common difference d is negative, the terms of the AP decrease.

Summary

Using the formula an = a + (n - 1)d, you can easily calculate any term in an arithmetic progression, knowing the first term, the common difference, and the desired term's position.

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