Question

For `U_n=int_0^1x^n(2-x)^n dx ; V_n=int_0^1x^n(1-x)^ndxn in N ,` which of the following statement(s) is/are true? `U_n=2^n V_n` (b) `U_n=2^(-n)V_n` `U_n=2^(2n)V_n` (d) `V_n=2^(-2n)U_n`

Answer 1

Correct Answer - NA
We have `U_(n)=int_(0)^(1)x^(n).(2-x)^(n)dx`
Put `x=2t`
`:. dx=2dt`
`:. U_(n)=2int_(0)^(1//2)2^(n).t^(n)2^(n)(1-t)^(n)dt`
`:.U_(n)=2^(2n+1)int_(0)^(1//2)x^(n)(1-x)^(n)dx`…………..1
Now `V_(n)=int_(0)^(1)x^(n)(1-x)^(n)dx`
`=2int_(0)^(1//2)x^(n)(1-x)^(n)dx`................2
From 1 and 2
`U_(n)=2^(2n).V_(n)`