If `S_(r) = sqrt(r+sqrt(r+sqrt(r+sqrt(r+.....oo)))),r gt 0,` then which of the following is/are correct.

Options

A. `S_(r),S_(6),S_(12),S_(20),` are in AP
B. `S_(4),S_(9),S_(16)` are irrational
C. `(2S_(4)-1)^(2),(2S_(5)-1)^(2),(2S_(6)-1)^(2)` are in AP
D. `S_(2),S_(12),S_(56)` are in GP

Answer

A, B, C, D

Explanation

`:.S_(r) = sqrt(r+ sqrt(r+sqrt(r+sqrt(......oo))))=sqrt(r+S)`
`S_(r)^(2)-S_(r)-r=0`
`:.S_(r)=(1+sqrt((1+4r)))/(2) " " [:.rgt0]S_(r)-r=0`
Alternate (a) `S_(2),S_(6),S_(12),S_(20)i.e.2,3,4,5` are in AP.
Alternate(b) `S_(4),S_(9),S_(16),i.e.(1+sqrt(17))/(2),(1+sqrt(37))/(2),(1+sqrt(65))/(2)` are irrationals.
Alternate (c) `(2S_(4-1))^(2),(2S_(5-1))^(2),(2S_(6-1))^(2)i.e.,17,21,25` are in AP.
Alternate (d) `S_(2),S_(12),S_(56)i.e.,2,4,8` are in GP.