To find the number of terms in an Arithmetic Progression (AP) in Class 10, you can use the formula for the nth term of an AP. An Arithmetic Progression is a sequence where the difference between consecutive terms is constant. This constant difference is called the common difference.
Understanding Arithmetic Progression (AP)
In an AP, we denote:
- a as the first term
- d as the common difference
- n as the number of terms
- an as the nth term (also known as the last term if we know the total number of terms)
Formula for the nth Term
The formula to find the nth term of an AP is given by:
an = a + (n - 1)d
This formula, derived from the reference material, is essential for finding the number of terms.
Finding the Number of Terms (n)
If you know the first term (a), the common difference (d), and the last term (an), you can rearrange the formula to solve for n:
- Start with the formula: an = a + (n - 1)d
- Isolate (n - 1)d: an - a = (n - 1)d
- Divide by d: (an - a) / d = n - 1
- Solve for n: n = (an - a) / d + 1
Therefore, the formula to calculate the number of terms (n) in an AP is:
n = (an - a) / d + 1
Example
Let's say you have an AP: 2, 5, 8, ..., 50. Find the number of terms.
- a (first term) = 2
- d (common difference) = 5 - 2 = 3
- an (last term) = 50
Using the formula:
n = (50 - 2) / 3 + 1
n = 48 / 3 + 1
n = 16 + 1
n = 17
Therefore, there are 17 terms in the AP.
Steps Summarized
Here's a step-by-step guide to finding the number of terms in an AP:
- Identify: Determine the first term (a), common difference (d), and the last term (an).
- Apply the Formula: Use the formula n = (an - a) / d + 1.
- Calculate: Substitute the values of a, d, and an into the formula and solve for n.
By following these steps and understanding the formula, you can easily find the number of terms in any given Arithmetic Progression.
