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Prove that C0+2C1+4C2+8C3+.....+2^nCn=3^...

Prove that `C_0+2C_1+4C_2+8C_3+.....+2^nC_n=3^n` .

Text Solution

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`S=C_(0)-2^(2)C_(1)+3^(2)C_(2)-"....."+(-1)^(n)(n+1)^(2)C_(n)`
`T_(r) = (-1)^(r)r^(2).^(n)C_(r)`
`= (-1)^(r)r(r^(n)C_(r))`
`= (-1)^(r)r(n^(n-1)C_(r-1))`
`=n(-1)^(r)((r-1)+1)(.^(n-1)C_(r-1))`
`=n(-1)^(r)((r-1).^(n-1)C_(r-1)+.^(n-1)C_(r-1))`
`= n(-1)^(r)((n-1)^(n-2)C_(r-2)+.^(n-1)C_(r-1))`
`= n(n-1).^(n-2)C_(r-2)(-1)^(r-2)-n^(n-1)C_(r-1)(-1)^(r-1)`
`rArr S = underset(r=0)overset(n)sumT_(r)`
`= n(n-1)(1-1)^(n-2)-n(1-1)^(n-1)`
`=0`
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