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Binomial Theorem(Part-6)

ByBasanta Kumar Khanal
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Basanta Kumar Khanal
Basanta Kumar Khanal
३२ असार २०८३

Middle Term(s) in a Binomial Expansion

The middle term(s) of a binomial expansion are the term(s) that lie at the center of the expanded expression. The number of middle terms depends on the total number of terms in the expansion.

For the expansion of (a+b)n(a+b)^n:

  • The total number of terms is n+1n+1n+1.
  • If nnn is even, there is one middle term.
  • If nnn is odd, there are two middle terms.

How to Find the Middle Term(s)

  • If n is even:
    The middle term is the (n2+1)th\left(\frac{n}{2}+1\right)^{\text{th}} term.
  • If n is odd:
    The middle terms are the (n+12)th\left(\frac{n+1}{2}\right)^{\text{th}} and (n+32)th\left(\frac{n+3}{2}\right)^{\text{th}} terms.

Examples

  • In (a+b)6(a+b)^6, there are 7 terms, so the 4th term is the middle term.
  • In (a+b)7(a+b)^7, there are 8 terms, so the 4th and 5th terms are the middle terms.

Importance

Finding the middle term(s) is useful in algebraic simplification, competitive examinations, and solving problems involving specific terms of a binomial expansion.

Conclusion

The concept of middle term(s) helps students identify central terms in a binomial expansion quickly without writing the complete expansion. It is an essential application of the Binomial Theorem and frequently appears in mathematical problem-solving.

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