hiiiiii

this is a very standard sort of question .. let us try to solve it ..

 

R=(55+11)²ⁿ⁺¹ and F=R-[R]

 

Now since F = R - [R] , R = F + [R]

 

so [R] + F = (55+11)²ⁿ⁺¹    , 0 <= F <1  

 

Now let F1 be (55-11)²ⁿ⁺¹ , clearly 0 < F1 < 1

 

So [R] + F - F1 = (55+11)²ⁿ⁺¹ - (55-11)²ⁿ⁺¹

                              = 2 { ²ⁿ⁺¹C₁(55)²ⁿ(11) + ²ⁿ⁺¹C₃(55)^2n-2(11)³ + ... }

                      = 2k , where k is an integer  

 

Therefore , F - F1 = 2k - [R]

 

Since 2k - [R] is integer , F - F1 is also integer .

 

But -1 < F - F1 < 1

 

so, F - F1 = 0  or F = F1

 

Thus RF = RF1 = (55+11)²ⁿ⁺¹(55-11)²ⁿ⁺¹

 

                       = (125 - 121)²ⁿ⁺¹

                              = (4)²ⁿ⁺¹ 

 

I hope the solution is clear ..

 

cheers