What is the number of terms in an arithmetic progression?

The number of terms in an arithmetic progression can be found by a straightforward calculation involving the first term, last term, and common difference.

Calculating the Number of Terms

According to the provided reference, to find the number of terms in an arithmetic progression:

1. Subtract the first term from the last term.
2. Divide the result by the common difference.
3. Add 1 to the quotient obtained in step 2.

This can be represented by the following formula:

Number of terms = (Last term - First term) / Common difference + 1

Formula:

n = (l - a) / d + 1

Where:

• n = number of terms
• l = last term
• a = first term
• d = common difference

Example

Let's take an example: Consider the arithmetic progression: 2, 5, 8, 11, 14.

• First term (a) = 2
• Last term (l) = 14
• Common difference (d) = 3

Number of terms = (14 - 2) / 3 + 1
Number of terms = 12 / 3 + 1
Number of terms = 4 + 1
Number of terms = 5

Therefore, the number of terms in this arithmetic progression is 5.

Key Concepts

• Arithmetic Progression: A sequence of numbers such that the difference between any two consecutive terms is constant.
• Common Difference: The constant difference between consecutive terms in an arithmetic progression.

This method allows for a simple and direct calculation of the number of terms, making it a fundamental concept in working with arithmetic progressions.