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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.

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Student-friendly explanation

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.

How to write this in exams

  1. 1

    Start with the exact idea

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

  2. 2

    Then show how to use it

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

  3. 3

    Add one concrete example

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

  4. 4

    Avoid this incomplete answer

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

Definition

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

Example

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

Rule to remember

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

Memory hook

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

Examples and method

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 to apply

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

Diagram support

A box or bracket showing the first n terms being added can help students understand why a sum formula is needed.

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.

Avoid common mistakes

Common confusion

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

Common wrong answer

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

Exam tip

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

Quick check

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

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.

Answer writing and exam use

1-mark answer

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

2-mark answer

The sum of first n terms of an AP is the total obtained by adding the first n terms of the sequence. S_n = n/2 [2a + (n - 1)d] or S_n = n/2(a + l), where l is the last term. For 1, 3, 5, 7, the sum of the first 4 terms is 16.

3-mark answer

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. S_n = n/2 [2a + (n - 1)d] or S_n = n/2(a + l), where l is the last term. For 2, 5, 8, 11, ... the sum of first 4 terms is S4 = 4/2 [2(2) + (4 - 1)3] = 2[4 + 9] = 26. Students may be asked for the sum of some terms, a total amount saved, or a total length based on an AP pattern. A common wrong answer is to use a_n instead of S_n, which gives only one term, not the total.
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