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  • In the expansion of  which of the following is always true?<div cl

In the expansion of \large (x+iy)^n which of the following is always true?

Option: 1

Middle term is real if n is even


Option: 2

Middle term is real if n is odd


Option: 3

Middle term is imaginary if n is even


Option: 4

None of the above


Answers (1)

best_answer

Middle Term

The middle term in the expansion (x + y)n, depends on the value of 'n'. 

Case 1 When 'n' is even

Only one middle term : \left(\frac{\mathrm{n}}{2}+1\right)^{\mathrm{th}}term.

It is given by \mathrm{T} _{\frac{\mathrm{n}}{2}+1}=\left(\begin{array}{c}{\mathrm{n}} \\ {\frac{\mathrm{n}}{2}}\end{array}\right) \mathrm{x}^{\frac{\mathrm{n}}{2}} \mathrm{y}^{\frac{\mathrm{n}}{2}} .

 

Case 2 When 'n' is odd

Two middle terms in the expansion, \left(\frac{n+1}{2}\right)^{t h} \text { and }\left(\frac{n+3}{2}\right)^{t h} term.

And it is given by

\mathrm{T}_{\frac{\mathrm{n}+1}{2}}=\left(\begin{array}{c}{\mathrm{n}} \\ {\frac{\mathrm{n}-1}{2}}\end{array}\right) \mathrm{x}^{\frac{\mathrm{n}+1}{2}} \cdot \mathrm{y}^{\frac{\mathrm{n}-1}{2}} \quad \,\,and\,\,\mathrm{T}_{\frac{\mathrm{n}+3}{2}}=\left(\begin{array}{c}{\mathrm{n}} \\ {\frac{\mathrm{n}+1}{2}}\end{array}\right) \mathrm{x}^{\frac{\mathrm{n}-1}{2}} \cdot \mathrm{y}^{\frac{\mathrm{n}+1}{2}}

 

Now,

 (x+iy)^n= \sum_{r=0}^{n} (^n C_{r} . (x)^{n-r} . (iy)^r)

Middle term of the given expansion if n is even

^nC_{n/2} (x)^{n-\frac{n}{2}}(iy)^{\frac{n}{2}}

The imaginary part is dependent on n/2, and n/2 can be even or odd because if n/2 is even then it is real and if n/2 is odd then it is imaginary

Hence option A and C is incorrect

 

Now if n is odd

Middle terms are \large \mathrm{T}_{\frac{\mathrm{n}+1}{2}}=\left(\begin{array}{c}{\mathrm{n}} \\ {\frac{\mathrm{n}-1}{2}}\end{array}\right) \mathrm{x}^{\frac{\mathrm{n}+1}{2}} \cdot \mathrm{(iy)}^{\frac{\mathrm{n}-1}{2}} \quad \mathrm{T}_{\frac{\mathrm{n}+3}{2}}=\left(\begin{array}{c}{\mathrm{n}} \\ {\frac{\mathrm{n}+1}{2}}\end{array}\right) \mathrm{x}^{\frac{\mathrm{n}-1}{2}} \cdot \mathrm{(iy)}^{\frac{\mathrm{n}+1}{2}}

Here imaginary part depends on (n-1)/2 and (n+1)/2 which can be odd or even

Hence none of the statement is always true.

Posted by

Rakesh

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