Understanding the Nth Term from the End of an AP
In an Arithmetic Progression (AP), the nth term from the end can be calculated using a straightforward formula. Here’s how you can derive it step by step:
Key Components of an AP
- Common Difference (d): The difference between consecutive terms in the sequence.
- Last Term (l): The final term of the AP.
- Number of Terms (n): The total number of terms in the AP, which can be determined using the formula:
n = (l - a) / d + 1, where 'a' is the first term.
Formula for the Nth Term from the End
To find the nth term from the end of the AP, use the formula:
Nth term from the end = l - (n - k) * d
Where:
- k: Represents the position from the end (e.g., k = 1 for the last term, k = 2 for the second last term, etc.).
Steps to Calculate
1. Identify the First Term (a): If not provided, use the context of the problem to determine it.
2. Calculate n: Use the formula provided above to find the total number of terms.
3. Substitute Values: Plug in the values of l, n, d, and k into the formula.
4. Solve for the Nth Term: Perform the arithmetic to find the specific term from the end.
Example
If you have an AP with a first term of 2, a common difference of 3, and a last term of 20, you can calculate the number of terms and then find any nth term from the end by following the steps outlined above.
This method will give you the correct term without confusion, ensuring clarity in your calculations.