To find the number of terms in an Arithmetic Progression (AP), you can use a specific formula derived from the properties of APs. This formula relates the last term, first term, common difference, and the number of terms.
Formula and Explanation
The formula to find the number of terms (n) in an AP is derived from the general term formula:
- tn = a + (n - 1) d
Where:
- tn is the last term of the AP.
- a is the first term of the AP.
- n is the number of terms in the AP (what we want to find).
- d is the common difference between consecutive terms.
To find n, you simply rearrange the formula and solve for it. According to the reference, you plug the given values into the formula tn = a + (n - 1) d and solve for n.
Steps to Find the Number of Terms
Here's a step-by-step guide on how to determine the number of terms in an AP:
- Identify the First Term (a): Determine the first term in the given AP.
- Identify the Last Term (tn): Determine the last term in the given AP.
- Calculate the Common Difference (d): Find the common difference by subtracting any term from its succeeding term (e.g., second term - first term).
- Apply the Formula: Substitute the values of tn, a, and d into the formula tn = a + (n - 1) d.
- Solve for n: Solve the equation for n to find the number of terms.
Example
Let's say we have an AP: 2, 5, 8, ..., 32.
- a (First term) = 2
- tn (Last term) = 32
- d (Common difference) = 5 - 2 = 3
Now, using the formula:
32 = 2 + (n - 1) * 3
32 = 2 + 3n - 3
32 = 3n - 1
33 = 3n
n = 11
Therefore, there are 11 terms in the AP.
Summary
| Element | Description |
|---|---|
| tn | Last term of the AP |
| a | First term of the AP |
| n | Number of terms in the AP (what you are solving for) |
| d | Common difference of the AP |
| Formula to find n | tn = a + (n - 1) d |
