Q. 474.7( 3 Votes )
Prove that the coefficient of xn in the binomial expansion of (1 + x)2n is twice the coefficient of xn in the binomial expansion of (1 + x)2n-1.
Answer :
To Prove : coefficient of xn in (1+x)2n = 2 × coefficient of xn in (1+x)2n-1
For (1+x)2n,
a=1, b=x and m=2n
We have a formula,
To get the coefficient of xn, we must have,
xn = xr
• r = n
Therefore, the coefficient of xn 
………
………..
………cancelling n
Therefore, the coefficient of xn in (1+x)2n
………eq(1)
Now for (1+x)2n-1,
a=1, b=x and m=2n-1
We have formula,
To get the coefficient of xn, we must have,
xn = xr
• r = n
Therefore, the coefficient of xn in (1+x)2n-1 
…..multiplying and dividing by 2
Therefore,
coefficient of xn in (1+x)2n-1 = � × coefficient of xn in (1+x)2n
or coefficient of xn in (1+x)2n = 2 × coefficient of xn in (1+x)2n-1
Hence proved.
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