1.

यदि `(1+x)^(n)=C_(0)+C_(1)x+C_(2)x^(2)+...+C_(n)x^(n)`, साबित कीजिए कि `C_(1)+2.C_(2)+3.C_(3)+…+n.C_(n)=n2^(n-1)`

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


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