sana.noor wrote:If n is a positive integer and Tn is the sum of all the positive integers from 1 to n inclusive, is n even?
(1) Tn is even
(2) T2n is even

NOTE: the n and 2n are supposed to be in subscript.

Target question: Is n even?

Given: Tn is the sum of all the positive integers from 1 to n inclusive
For example T3 (T subscript 3) = 1+2+3 = 6
Similarly, T7 = 1+2+3+4+5+6+7 = 28

Statement 1: Tn is even
There are several values of n that meet this condition. Here are two:
Case a: n = 3 (since T3 = 6, which is even). In this case, n is odd
Case b: n = 4 (since T4 = 10, which is even). In this case, n is even
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: T2n is even
There's a nice formula for finding sums of integers from 1 to some value:
The sum of integers from 1 to k = k(k+1)/2

Let's apply the above formula to the sum of numbers from 1 to 2n:
So, T2n = (2n)(2n+1)/2
= n(2n+1)

IMPORTANT: Statement 2 tells us that T2n is even.
This means that n(2n+1) is even.
Now, (2n+1) is an odd value for all integer values of n (in fact this is the definition of an odd number)
So, if (2n+1) is odd, and n(2n+1) is even, then it must be the case that n is even.
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Answer = B

Cheers,
Brent