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Question

Show that the coefficient of the middle term of (1+x)2n is equal to the sum of the coefficients of the two middle terms of (1+x)2n1

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Solution

Mid terms of (1+x)2x
T2n+22=Tn+1=2nCn+1
Mid term of (1+x)2n1
T2n1+12 and T2n1+32
Tn and Tn+1
Tn=2n1Cn and Tn+1=2n1Cn+1
Tn+Tn+1=2n1Cn+2n1Cn+1
We know that:-
n1Cr+n1Cr+1=nCr+1
2n1Cn+2n1Cn+1
=2nCn+1

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