Understanding the Formula

How to Find the nth Term of an Arithmetic Progression?

Finding the nth term of an arithmetic progression (AP) is straightforward using a simple formula. An arithmetic progression is a sequence of numbers where the difference between consecutive terms remains constant. This constant difference is called the common difference, often denoted as d.

The formula to calculate the nth term (a_n) of an arithmetic progression is:

a_n = a₁ + (n - 1)d

Where:

• a_n is the nth term of the sequence.
• a₁ is the first term of the sequence.
• n is the position of the term you want to find (e.g., 5th term, 10th term).
• d is the common difference between consecutive terms.

Step-by-Step Calculation

To find the nth term:

1. Identify the first term (a₁): This is the starting number in the sequence.
2. Determine the common difference (d): Subtract any term from the term that follows it. The result will always be the same in an arithmetic progression.
3. Substitute the values into the formula: Plug in the values for a₁, n, and d into the formula a<sub>n</sub> = a<sub>1</sub> + (n - 1)d.
4. Calculate: Perform the arithmetic operations to find the value of a_n.

Examples

Let's illustrate with a couple of examples:

Example 1:

Find the 10th term of the arithmetic sequence: 2, 6, 10, 14…

• a₁ = 2
• d = 6 - 2 = 4
• n = 10

Using the formula: a₁₀ = 2 + (10 - 1) * 4 = 2 + 36 = 38

Therefore, the 10th term is 38.

Example 2:

An arithmetic sequence has a first term of 5 and a common difference of 4. What is the 7th term?

• a₁ = 5
• d = 4
• n = 7

Using the formula: a₇ = 5 + (7 - 1) * 4 = 5 + 24 = 29

The 7th term is 29.

Several online resources, including YouTube videos (Learn How to Find the nth Term of an Arithmetic Sequence ... - YouTube, Learn How to Find the nth Term of an Arithmetic Sequence Example ...) and websites like Cuemath (How to find the nth Term of an Arithmetic Sequence? [Solved]), provide further explanations and examples.