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Middle term in an AP

The middle term of an AP is the term that lies exactly between two equal numbers of terms on both sides.

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Student-friendly explanation

When the number of terms in an AP is odd, the middle term is the term at the center. In a well-balanced AP, the middle term is also the average of the first and last terms when the terms are arranged symmetrically.

How to write this in exams

  1. 1

    Start with the exact idea

    The middle term of an AP is the term that lies exactly between two equal numbers of terms on both sides.

  2. 2

    Then show how to use it

    1. Count the total terms. 2. Check whether the count is odd. 3. Find the central position using (n + 1)/2. 4. Read that term from the AP.

  3. 3

    Add one concrete example

    In 2, 5, 8, 11, 14, the middle term is 8.

  4. 4

    Avoid this incomplete answer

    A common wrong answer is to pick the term in the center by appearance without counting all the terms carefully.

Definition

The middle term of an AP is the term that lies exactly between two equal numbers of terms on both sides.

Example

In 2, 5, 8, 11, 14, the middle term is 8.

Rule to remember

For an AP with an odd number of terms, middle term = a_(n+1)/2.

Memory hook

Middle means same number of terms on both sides.

Examples and method

Worked example

In 4, 7, 10, 13, 16, 19, 22, there are 7 terms, so the 4th term is the middle term, which is 13.

Method to apply

1. Count the total terms. 2. Check whether the count is odd. 3. Find the central position using (n + 1)/2. 4. Read that term from the AP.

Diagram support

Write the terms in a row and mark equal terms on both sides of the center.

How CBSE asks it

Students may be asked to identify the central term or use the middle term in relation to the first and last terms.

Avoid common mistakes

Common confusion

Students sometimes call the third term the middle term without checking whether the sequence has an odd number of terms.

Common wrong answer

A common wrong answer is to pick the term in the center by appearance without counting all the terms carefully.

Exam tip

First count the total number of terms. Only then decide which one is in the middle.

Quick check

How do you find the middle term of an AP with an odd number of terms?

Count the total number of terms and choose the central term. For an odd number of terms, the middle term is the one with equal terms on both sides.

Answer writing and exam use

1-mark answer

The middle term of an AP is the term that lies exactly between two equal numbers of terms on both sides.

2-mark answer

The middle term of an AP is the term that lies exactly between two equal numbers of terms on both sides. For an AP with an odd number of terms, middle term = a_(n+1)/2. In 2, 5, 8, 11, 14, the middle term is 8.

3-mark answer

When the number of terms in an AP is odd, the middle term is the term at the center. In a well-balanced AP, the middle term is also the average of the first and last terms when the terms are arranged symmetrically. For an AP with an odd number of terms, middle term = a_(n+1)/2. In 4, 7, 10, 13, 16, 19, 22, there are 7 terms, so the 4th term is the middle term, which is 13. Students may be asked to identify the central term or use the middle term in relation to the first and last terms. A common wrong answer is to pick the term in the center by appearance without counting all the terms carefully.
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