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Nth term of an AP

The nth term of an AP is the formula used to find any term directly without writing all the earlier terms.

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Student-friendly explanation

The nth term formula saves time in exams because it gives the required term in one step. For an AP with first term a and common difference d, the nth term is found by starting from the first term and adding the common difference one less time than the term number.

How to write this in exams

  1. 1

    Start with the exact idea

    The nth term of an AP is the formula used to find any term directly without writing all the earlier terms.

  2. 2

    Then show how to use it

    1. Identify the first term a. 2. Find d. 3. Put the required n into a_n = a + (n - 1)d. 4. Simplify carefully.

  3. 3

    Add one concrete example

    For 5, 9, 13, 17, ... the 4th term is 5 + 3(4 - 1) = 14.

  4. 4

    Avoid this incomplete answer

    A common wrong answer is a + nd. That adds one extra difference and shifts every term forward.

Definition

The nth term of an AP is the formula used to find any term directly without writing all the earlier terms.

Example

For 5, 9, 13, 17, ... the 4th term is 5 + 3(4 - 1) = 14.

Rule to remember

a_n = a + (n - 1)d, where a is the first term, d is the common difference, and n is the term number.

Memory hook

First term plus one less set of jumps than the term number.

Examples and method

Worked example

For the AP 2, 7, 12, 17, ... we have a = 2 and d = 5. The 6th term is a6 = 2 + (6 - 1)5 = 2 + 25 = 27.

Method to apply

1. Identify the first term a. 2. Find d. 3. Put the required n into a_n = a + (n - 1)d. 4. Simplify carefully.

Diagram support

A term ladder with labels a, a + d, a + 2d, a + 3d helps show why the formula uses n - 1 differences.

How CBSE asks it

CBSE often asks for a particular term, the position of a term, or a term in a sequence given as an AP.

Avoid common mistakes

Common confusion

Students often use n instead of n - 1, which gives one extra common difference.

Common wrong answer

A common wrong answer is a + nd. That adds one extra difference and shifts every term forward.

Exam tip

Write the formula first, then substitute a, d, and n carefully before simplifying.

Quick check

What formula is used to find the nth term of an AP?

The nth term of an AP is a_n = a + (n - 1)d, where a is the first term and d is the common difference. This formula helps you find any term directly.

Answer writing and exam use

1-mark answer

The nth term of an AP is the formula used to find any term directly without writing all the earlier terms.

2-mark answer

The nth term of an AP is the formula used to find any term directly without writing all the earlier terms. a_n = a + (n - 1)d, where a is the first term, d is the common difference, and n is the term number. For 5, 9, 13, 17, ... the 4th term is 5 + 3(4 - 1) = 14.

3-mark answer

The nth term formula saves time in exams because it gives the required term in one step. For an AP with first term a and common difference d, the nth term is found by starting from the first term and adding the common difference one less time than the term number. a_n = a + (n - 1)d, where a is the first term, d is the common difference, and n is the term number. For the AP 2, 7, 12, 17, ... we have a = 2 and d = 5. The 6th term is a6 = 2 + (6 - 1)5 = 2 + 25 = 27. CBSE often asks for a particular term, the position of a term, or a term in a sequence given as an AP. A common wrong answer is a + nd. That adds one extra difference and shifts every term forward.
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