Let `f_(n)(theta) = sum_(n=0)^(n) (1)/(4^(n))sin^(4)(2^(n)theta)`. Then which of the following alternative(s) is/are correct ?

Options

A. `f_(2)((pi)/(4))=(pi)/(sqrt(2))`
B. `f_(3)((pi)/(8)) = (2 + sqrt(2))/(4)`
C. `f_(4)((3pi)/(2)) = 1`
D. `f_(5)(pi) = 0`

Answer

C, D

Explanation

`f_(n)(theta)=overset(n)underset(n=0)(sum)(1)/(4^(n))sin^(4)(2^(n)theta)= overset(n)underset(n=0)sum(1)/(4^(n))(sin^(2)(2^(n)0))(1-cos^(2)2^(n)theta)`
`=overset(n)underset(n=0)(sum)[sin^(2)(2^(n)theta)-(1)/(4)sin^(2)(2^(n-1)theta)](1)/(4^(n))`
`rArrf_(n)(theta)overset(n)underset(n=0)(sum)[(1)/(4^(n))sin^(2)2^(n)theta-(1)/(4^(n+1))sin^(2)(2^(n+1)theta)]`
`:. f_(n)(theta) = (sin^(2) 2theta- (1)/(4)sin^(2)2theta)`
`+((1)/(4)sin^(2)2theta-(1)/(4^(2))sin^(2)2^(2)theta)+.......(n)` term
`f_(n)(theta) = sin^(2)theta - (1)/(4^(n-1))sin^(2)(2^(n+1)theta)`