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)2n−1
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Solution
⟹ Mid terms of (1+x)2x T2n+22=Tn+1=2nCn+1 Mid term of (1+x)2n−1 T2n−1+12 and T2n−1+32 Tn and Tn+1 Tn=2n−1Cn and Tn+1=2n−1Cn+1 Tn+Tn+1=2n−1Cn+2n−1Cn+1 We know that:- n−1Cr+n−1Cr+1=nCr+1 ∴2n−1Cn+2n−1Cn+1 =2nCn+1