Question

Find the middle terms in the expansion of \(\rm \left(x + \frac 1 x \right)^{10}\)
1. ¹⁰C₄
2. ¹⁰C₆
3. ¹⁰C₅
4. ¹⁰C₇

Answer 1

Correct Answer - Option 3 : ¹⁰C₅

Concept:

General term: General term in the expansion of (x + y)n is given by

\(\rm {T_{\left( {r\; + \;1} \right)}} = \;{\;^n}{C_r} × {x^{n - r}} × {y^r}\)

 

Middle terms: The middle terms is the expansion of (x + y) n depends upon the value of n.

• If n is even, then there is only one middle term i.e. \(\rm \left( {\frac{n}{2} + 1} \right){{\rm{\;}}^{th}}\) term is the middle term.
• If n is odd, then there are two middle terms i.e. \(\rm {\left( {\frac{{n\; + \;1}}{2}} \right)^{th}}\;\)and \(\rm {\left( {\frac{{n\; + \;3}}{2}} \right)^{th}}\) are two middle terms.

 

 

Calculation:

Here, we have to find the middle terms in the expansion of \(\rm \left(x + \frac 1 x \right)^{10}\)

Here n = 10 (n is even number)

∴ Middle term = \(\rm \left( {\frac{n}{2} + 1} \right) = \left( {\frac{10}{2} + 1} \right) =6th\;term\)

T₆ = T (5 + 1) = ¹⁰C₅ × (x) (10 - 5) × \(\rm \left(\frac {1}{x}\right)^5\)

T₆ =  ¹⁰C5