The value of lim _(n → ∞) ([r] + [2r] + ... + [nr])/n², where r is a non-zero real number and [r] denotes the greatest integer less than or equal to r, is equal to :

Solution

Maths sir · 4 steps

Trap each [kr] between kr - 1 and kr, add them up, and both ends squeeze to r/2.

1. Bound one term. For any real x, x - 1 < [x] ≤ x. With x = kr this gives kr - 1 < [kr] ≤ kr, and it holds for negative r too.
2. Add them up. Add these inequalities for k = 1 to n. This uses 1 + 2 + … + n = n(n+1)/2, and the -1 appears n times. Call the sum Sₙ.

   (r n(n+1))/2 - n < Sₙ ≤ (r n(n+1))/2

3. Divide by n squared. The right side becomes r/2(1 + 1/n) → r/2. The left side is the same thing minus 1/n, which also tends to r/2.
4. So. By the sandwich theorem, Sₙ/n² → r/2, which is option (b).

The idea to keep. Whenever greatest integer terms sit inside a limit, replace [x] by the bounds x - 1 < [x] ≤ x. The -1 errors add up to only about n, which dies against n².

Answer from the NTA final answer key. Solution by PrepWiser.