11make use of the method of mathematical induction in proving the given expression
If (1+x)^{n}=C_{0}+C_{1}x+C_{2}x^{2}+.......C_{n}x^{n}
then prove that \sum_{0\leq i}^{}{\sum_{<j\leq n }^{}{}}C_{i}C_{j}(i+j)=n(2^{2n-1}-\frac{1}{2}. ^{2n}C_{n})
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3 Answers
11
11 Σ Σ CiCj = (22n-(2nCn))/2....(1)
0≤i<j≤n
let P=ΣΣ(i+j)CiCj
replacing i by n-i and j by n-j,we get
P=ΣΣ(n-i+n-j)Cn-iCn-j
P=ΣΣ(2n-(i+j))CiCj
P=2nΣΣCiCj - ΣΣ(i+j)CiCj
but ΣΣ(i+j)CiCj = P
therefore,
2p = the expression given in (1)