Let R=(5 √5+11)2 n+1 and f=R-[R] where [.] denotes the greatest integer function then value of R f is

Question

Let R=(5 √5+11)2 n+1 and f=R-[R] where [.] denotes the greatest integer function then value of R f is (A) 22 n+1 (B) 42 n+1 (C) 24 n+1 (D) 42 n

Tardigrade Admin · Accepted Answer

R=(5 √5+11)2 n+1=1+f, 0 < f < 1 Let f prime=(5 √5-11)2 n+1, 0 < f prime < 1 ∴(5 √5)2 n+1+ 2 n+1 C1(5 √5)2 n .11+ ldots=I+f ∵-1 < f-f prime < 1 ∴(5 √5)2 n+1- 2 n+1 C1(5 √5)2 n ⋅ 11+ ldots . f prime ∴ 2( 2 n+1 C1 ⋅(√5 5)2 n ⋅ 11+ ldots.)=I+f-f prime=I ∴ R f=R f prime=(5 √5+11)2 n+1 ⋅(5 √5-11)2 n+1 =(125-121)2 n+1=42 n+1

Q. Let $R = (55 + 11)^{2 n + 1}$ and $f = R - [R]$ where [.] denotes the greatest integer function then value of $R f$ is

$2^{2 n + 1}$

$4^{2 n + 1}$

$2^{4 n + 1}$

$4^{2 n}$

Q. Let R=(55​+11)2n+1 and f=R−[R] where [.] denotes the greatest integer function then value of Rf is

22n+1

42n+1

24n+1

42n

R=(55​+11)2n+1=1+f,0<f<1 Let f′=(55​−11)2n+1,0<f′<1 ∴(55​)2n+1+2n+1C1​(55​)2n.11+…=I+f ∵−1<f−f′<1 ∴(55​)2n+1−2n+1C1​(55​)2n⋅11+….f′ ∴2(2n+1C1​⋅(5​5)2n⋅11+….)=I+f−f′=I ∴Rf=Rf′=(55​+11)2n+1⋅(55​−11)2n+1 =(125−121)2n+1=42n+1

Q. Let $R = (55 + 11)^{2 n + 1}$ and $f = R - [R]$ where [.] denotes the greatest integer function then value of $R f$ is

$2^{2 n + 1}$

$4^{2 n + 1}$

$2^{4 n + 1}$

$4^{2 n}$