Question

There are rooms numbered from 1 to 99 in an apartment. A mathematician notices that the sum of all room numbers before his room is equal to the sum of all room numbers after his room. Find his room number.

• A. 20
• B. 35
• C. 49
• D. 50

Hint:

This type of problem can be solved quickly with an alternative setup. Let (S_{before} = S_{after} = K). The total sum is (S_N = S_{before} + x + S_{after} = K + x + K = 2K + x). Substituting (K = S_{x-1}), we get (S_N = 2S_{x-1} + x). [ frac{N(N+1)}{2} = 2 frac{(x-1)x}{2} + x = x^2 - x + x = x^2 ] This gives the relation (x^2 = frac{N(N+1)}{2}). For an integer solution for (x), (frac{N(N+1)}{2}) must be a perfect square. For (N=49), (frac{49 times 50}{2} = 1225 = 35^2). This confirms (x=35).

Answer 1

The Correct Option is B