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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.

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Student-friendly explanation

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).

How to write this in exams

  1. 1

    Start with the exact idea

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

  2. 2

    Then show how to use it

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

  3. 3

    Add one concrete example

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

  4. 4

    Avoid this incomplete answer

    For first 10 terms of 4, 7, 10, ..., using l = 10 is wrong because 10 is the third term, not the tenth term.

Definition

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

Example

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

Rule to remember

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

Memory hook

AP sum pairs the beginning and ending terms.

Examples and method

Worked example

Find the sum of first 10 terms of AP 4, 7, 10, ... . Here a = 4, d = 3, n = 10. S10 = 10/2 × [2(4) + 9(3)] = 5 × 35 = 175.

Method to apply

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

Diagram support

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

How CBSE asks it

Questions may ask for total savings, total distance, seating arrangements, or direct sum of n AP terms.

Avoid common mistakes

Common confusion

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

Common wrong answer

For first 10 terms of 4, 7, 10, ..., using l = 10 is wrong because 10 is the third term, not the tenth term.

Exam tip

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

Quick check

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

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

Study the sum of first n terms of an ap diagram carefully

Use the labelled diagram to keep sum of first n terms of an ap clear in short answers and revision.

What this diagram makes clear

This diagram keeps the labels and direction of sum of first n terms of an ap in the right order.

Where this helps in exams

Use this for labelled diagram work and short exam answers on sum of first n terms of an ap.

Revision cue

Revise sum of first n terms of an ap through the labels before writing the answer.

Answer writing and exam use

1-mark answer

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

2-mark answer

The sum of the first n terms of an arithmetic progression is the total obtained by adding its first n terms. Sn = n/2 × (2a + (n - 1)d) and Sn = n/2 × (a + l), where l is the nth or last required term. For 2, 5, 8, 11, the sum of first 4 terms is 2 + 5 + 8 + 11 = 26.

3-mark answer

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). Sn = n/2 × (2a + (n - 1)d) and Sn = n/2 × (a + l), where l is the nth or last required term. Find the sum of first 10 terms of AP 4, 7, 10, ... . Here a = 4, d = 3, n = 10. S10 = 10/2 × [2(4) + 9(3)] = 5 × 35 = 175. Questions may ask for total savings, total distance, seating arrangements, or direct sum of n AP terms. For first 10 terms of 4, 7, 10, ..., using l = 10 is wrong because 10 is the third term, not the tenth term.
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