1.

यदि `(1+x)^(n)=C_(0)+C_(1)x+C_(2)x^(2)+...+C_(n)x^(n)`, साबित कीजिए कि `C_(0)+2.C_(1)+3.C_(2)+…+(n+1).C_(n)=2^(n-1)(n+2)`

Answer» दिया गया श्रेणी है : `C_(0)+2.C_(1)+3.C_(2)+...+(n+1).C_(n)`
`=[(r-1)+1].""^(n)C_(r-1)=(r-1).""^(n)C_(r-1)+""^(n)C_(r-1)`
`=n.""^(n-1)C_(r-2)+""^(n)C_(r-1)" "[because (r-1).""^(n)C_(r-1)=n.""^(n-1)C_(r-2)]`
अब `" "C_(0)+2.C_(1)+3.C_(2)+.....+(n+1).C_(n)=underset(r=1)overset(n+1)sum t_(r)`
`=underset(r=1)overset(n+1)sumn.""^(n-1)C_(r-2)+underset(r=1)overset(n+1)sum""^(n)C_(r-1)=n underset(r=1)overset(n+1)sum""^(n-1)C_(r-2)+underset(r=1)overset(n+1)""^(n)C_(r-1)`
`=n[""^(n-1)C_(0)+""^(n-1)C_(1)+...+""^(n-1)C_(n-1)]+(""^(n)C_(0)+""^(n)C_(1)+...+""^(n)C_(n))`
`=n.2^(n-1)+2^(n)=2^(n-1)(n+2)`


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