The sum of n terms of an Arithmetic Progression (AP) helps us add the first n numbers of a sequence without manually adding each term. This formula is extremely useful for long sequences where adding terms one by one becomes time-consuming. Whether you're solving exam problems or calculating totals in real life, this AP sum formula makes the process faster and accurate.
Formula for Sum of n Terms of an AP – Overview
| Formula | Variables | When it is Used |
|---|---|---|
| Sn = n/2 [2a + (n – 1)d] | Sn = sum of n terms | Used to calculate total of AP terms |
| a = first term | Helpful in exams and financial calculations | |
| d = common difference | Used for series with regular increments | |
| n = number of terms |
What is Sum of n Terms in Maths?
The sum of n terms in an Arithmetic Progression means adding the first n numbers of a sequence that increases or decreases by a fixed difference. For example, in 2, 4, 6, 8, the sum of the first 4 terms is 20.
In AP, each term changes by a constant difference, making calculations simple. Instead of adding every term manually, we use the formula:
Sn = n/2 [2a + (n – 1)d]
Steps:
- Identify the first term (a), common difference (d), and number of terms (n).
- Substitute these values into the formula.
- Solve step by step to get the total.
This formula is widely used in CUET, SSC, Banking, Railways, JEE, and in real-life calculations such as savings, installment totals, or sequences that grow evenly.
Examples to Calculate Sum of n Terms of AP
Example 1: AP = 2, 5, 8 … (Find sum of first 10 terms)
Step 1: a = 2, d = 3, n = 10
Step 2: Sn = 10/2 [2×2 + (10 – 1)×3]
= 5 [4 + 27]
= 5 × 31
Result: 155
So, the sum of the first 10 terms is 155.
Example 2: AP = 7, 14, 21 … (Find sum of first 15 terms)
Step 1: a = 7, d = 7, n = 15
Step 2: Sn = 15/2 [2×7 + (15 – 1)×7]
= 15/2 [14 + 98]
= 15/2 × 112
Result: 840
So, the sum of the first 15 terms is 840.
FAQs about Sum of n Terms Formula
Q1. Can AP have negative terms in the sum?
Yes, if the sequence has negative terms, the sum can also be negative.
Q2. What is the sum of first n natural numbers using AP?
Using a = 1 and d = 1, the formula becomes Sn = n(n + 1)/2.
Q3. Is AP used in competitive exams?
Yes, it is frequently asked in CUET, SSC, Banking, JEE, and Railways exams.
Q4. Can we find the sum if the last term is known instead of d?
Yes, you can use Sn = n/2 (a + l), where l is the last term.
Q5. Who introduced the AP sum formula?
It is associated with ancient mathematicians, but Gauss famously applied it for quick calculations.
Q6. Is AP only about increasing numbers?
No, an AP can also decrease if the common difference is negative.