1. Introduction
The sum of the first n terms of an AP tells us how much total value we get when we add the first n terms of the sequence.
Example: For the AP 3, 6, 9, 12, ... the sum of the first 4 terms is:
\(3 + 6 + 9 + 12 = 30\)
Instead of adding each term, we use formulas to find the sum quickly.
2. Sum of First n Terms Formula
There are two main formulas to find the sum of an AP. You can use either depending on the information given.
2.1. Formula 1: Using a, d, and n
\(S_n = \dfrac{n}{2} [2a + (n - 1)d]\)
This formula is useful when you know the first term \(a\), the common difference \(d\), and the number of terms \(n\).
2.2. Formula 2: Using a, l, and n
\(S_n = \dfrac{n}{2} (a + l)\)
Here, l is the last term of the AP. This formula is faster when the last term is given.
3. Why the Sum Formula Works (Simple Understanding)
If you write an AP forward and backward and add them, each pair of terms gives the same total.
Example AP: 2, 5, 8, 11
Forward: 2, 5, 8, 11
Backward: 11, 8, 5, 2
Each pair adds to 13.
There are n such pairs, so the total is:
\(2S_n = n(a + l)\)
→ \(S_n = \dfrac{n}{2}(a + l)\)
4. Worked Examples
Let’s understand the formulas with examples.
4.1. Example 1: Using a and d
AP: 4, 7, 10, 13, ...
Find the sum of the first 10 terms.
Here, \(a = 4\), \(d = 3\), \(n = 10\)
\(S_{10} = \dfrac{10}{2} [2(4) + 9(3)]\)
\(= 5 (8 + 27) = 5 × 35 = 175\)
4.2. Example 2: Using last term
AP: 6, 12, 18, 24, ...
Find the sum of the first 8 terms.
First term \(a = 6\)
Common difference \(d = 6\)
Last term \(l = a + (n-1)d = 6 + 7(6) = 48\)
\(S_8 = \dfrac{8}{2}(6 + 48) = 4 × 54 = 216\)
4.3. Example 3: Word Problem
A person saves Rs. 50 in the first month, Rs. 60 in the second, Rs. 70 in the third, and so on. How much does he save in 12 months?
This is an AP with:
\(a = 50\), \(d = 10\), \(n = 12\)
\(S_{12} = \dfrac{12}{2} [2(50) + 11(10)]\)
\(= 6 (100 + 110) = 6 × 210 = 1260\)
5. Using the Formula to Find n
Sometimes we know the sum and need to find how many terms are taken.
Example: The sum of the first n terms of an AP is 135, where \(a = 3\), \(d = 2\). Find n.
\(135 = \dfrac{n}{2} [2(3) + (n - 1)2]\)
→ This becomes a quadratic equation in n. Solve to find n.
6. Difference Between nth Term and Sum
nth Term: Tells the value of a specific term.
Sum of n Terms: Tells the total of the first n terms.
nth term → value.
Sum → total.
7. Common Mistakes
- Using wrong formula (mixing nth term and sum formula).
- Forgetting to multiply n/2 correctly.
- Incorrectly identifying the last term.
- Not applying brackets properly when substituting.
- Arithmetic mistakes with negative or large values.
8. Quick Practice
Solve the following:
- Find \(S_{20}\) for AP 5, 9, 13, 17, ...
- The 10th term of an AP is 45 and the first term is 5. Find the sum of the first 10 terms.
- If the sum of first n terms of an AP is 210 and a = 7, d = 3, find n.
- Find the sum of the first 15 multiples of 6.
9. Summary
- Use \(S_n = \dfrac{n}{2} [2a + (n - 1)d]\) when a and d are known.
- Use \(S_n = \dfrac{n}{2}(a + l)\) when the last term is known.
- Sum helps calculate total of repeated patterns like savings, distances, or salaries.
- Be careful with signs and arithmetic when substituting values.