IIT JEE , AIEEE Forum - Mathematics , Algebra

Pavankumar

If [X] denotes the greatest integer less than or equal to X and F=R-[R] where

R=(5

5+11)²ⁿ⁺¹then RF=_______ .

puneet

hiiiiii

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

R=(5

5+11)²ⁿ⁺¹and F=R-[R]

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

so [R] + F = (5

5+11)²ⁿ⁺¹, 0 <= F <1

Now let F1 be (5

5-11)²ⁿ⁺¹ , clearly 0 < F1 < 1

So [R] + F - F1 = (5

5+11)²ⁿ⁺¹-(5

5-11)²ⁿ⁺¹

= 2 { ²ⁿ⁺¹C₁(5

5)²ⁿ(11) + ²ⁿ⁺¹C₃(5

5)^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 = (5

5+11)²ⁿ⁺¹(5

5-11)²ⁿ⁺¹

= (125 - 121)²ⁿ⁺¹

= (4)²ⁿ⁺¹

I hope the solution is clear ..

cheers