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medium difficulty5 min

Sum of first n terms

The sum of first n terms of an AP is the total obtained by adding the first n terms of the sequence.

In one line: Sum means add all the required terms, not just the last one.

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Understand it

This idea helps in questions where a total amount, total score, total money, or total objects is asked. The sum formula is powerful because it avoids writing all terms one by one. Students must remember that the formula uses the first term, the common difference, and the number of terms.

Formula and rule

Sn = n/2 [2a + (n - 1)d] or Sn = n/2(a + l), where l is the last term.

See it worked out

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

Worked example

For 2, 5, 8, 11, ... the sum of first 4 terms is S4 = 4/2 [2(2) + (4 - 1)3] = 2[4 + 9] = 26.

Method you can reuse

  1. 1Identify a, d, and n.
  2. 2Choose the sum formula.
  3. 3Substitute carefully.
  4. 4Simplify step by step and check arithmetic.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

What is the difference between the nth term and the sum of first n terms?

Think first, then show answer
The nth term gives one specific term of the AP, while the sum of first n terms gives the total of the first n terms added together. They are used for different types of questions.

Common wrong answer

A common wrong answer is to use an instead of Sn, which gives only one term, not the total.

Don't mix this up

Students often mix up the formulas for nth term and sum of n terms.

Exam tip

Before solving, ask yourself whether the question wants one term or the total of many terms.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Students may be asked for the sum of some terms, a total amount saved, or a total length based on an AP pattern.
  1. 1

    Enough for 1 mark

    The sum of first n terms of an AP is the total obtained by adding the first n terms of the sequence.

  2. 2

    Add for 2 marks

    Example: For 1, 3, 5, 7, the sum of the first 4 terms is 16.

  3. 3

    Add for 3 marks

    The sum formula is powerful because it avoids writing all terms one by one.

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