medium difficulty8 min

Sum of First n Terms of an AP

The sum of the first n terms of an arithmetic progression is the total obtained by adding its first n terms.

In one line: AP sum pairs the beginning and ending terms.

Understand it

Instead of adding many terms one by one, we use a formula. If the first term is a and common difference is d, then Sn = n/2 × (2a + (n - 1)d). If the first and last terms are known, use Sn = n/2 × (a + l).

Example

For 2, 5, 8, 11, the sum of first 4 terms is 2 + 5 + 8 + 11 = 26.

Formula and rule

Sn = n/2 × (2a + (n - 1)d) and Sn = n/2 × (a + l), where l is the nth or last required term.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Find the sum of first 10 terms of AP 4, 7, 10, ... .

1Here a = 4, d = 3, n = 10.
2S₁₀ = 10/2 × [2(4) + 9(3)] = 5 × 35 = 175.

Method you can reuse

1Identify a, d, and n.
2Decide whether l is known.
3Select the suitable formula.
4Substitute carefully.
5Simplify bracket first, then multiply by n/2.

Visual tip

Pairing terms from the start and end can be shown by arrows: first with last, second with second-last.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Which AP sum formula is convenient when the first and last terms are already known?

Think first, then show answer

Sn = n/2 × (a + l) is convenient when the first term, last term, and number of terms are known.

Common wrong answer

For first 10 terms of 4, 7, 10, ..., using l = 10.

Why it loses marks: 10 is the third term, not the tenth term.

Don't mix this up

Students sometimes use the last visible term as l even when the question asks for more terms than shown.

Exam tip

Choose Sn = n/2 × (a + l) only when the last required term l is known or can be found correctly.