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If C(r ) = (n!)/(r!(n - r)!), then prove...

If `C_(r ) = (n!)/(r!(n - r)!)`, then prove that
`sqrt(C_(1)) + sqrt(C_(2)) + …. + sqrt(C_(n)) sqrt(n(2^(n) - 1))`

Text Solution

Verified by Experts

A.M. of `(1//2)th` powers `lt (1//2)th` power of A.M.
`therefore ((C_1)^((1)/(2))+(C_2)^((1)/(2))+...+(C_n)^((1)/(2)))/(n)lt ((C_1+C_2+....C_n)/(n))^(1//2)`
or ` (sqrt(C_1)+sqrt(C_2)+....+ sqrt(C_n))/(n) lt ((2^n-1)/(n))^(1//2)`
or ` sqrt(C_1)+sqrt(C_2)+....+sqrt(C_n)lt (n sqrt((2^n-1)))/(sqrt(n))`
Hence,
` sqrt(C_1)+sqrt(C_2)+....+sqrt(C_n) lt sqrt([n(2^n-1)])`
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