Let R = (5√5 + 11)2n+1 and let f be the fractional part of R. Then Rf is equal…

Correct answer: 42n+1

Let α = 5√5 + 11 and β = 11 - 5√5. Then αβ = 121 - 125 = -4.

For k = 2n + 1, the conjugate sum αk + βk is an integer. Also, |β| < 1 and β2n+1 is negative.

So the fractional part of R = α2n+1 is f = -β2n+1.

Therefore, Rf = α2n+1 x (-β2n+1) = -(αβ)2n+1 = -(-4)2n+1 = 42n+1.