Question

Find the sum of the odd numbers between 0 and 50.

• A. 635
• B. 675
• C. 625
• D. 650

Hint:

There are exactly 50 integers from 1 to 50, of which exactly half (25) are odd.
The sum of the first \(n\) odd numbers is always \(n^2\).
Thus, the sum is simply \(25^2 = 625\). This is a standard identity worth memorizing.

Answer 1

The Correct Option is C

Step 1: List the sequence and count the terms.
The odd numbers between 0 and 50 are $1, 3, 5, \dots, 49$, which gives $n = 25$ terms in total.
Step 2: Pair them from the two ends.
Pairing the first term with the last, the second with the second-last, and so on, each pair adds up to the same total: \[ 1 + 49 = 50,\quad 3 + 47 = 50,\quad 5+45 = 50, \ \dots \] With 25 terms (an odd count), we get 12 full pairs of 50 plus one term left over in the middle, which is the 13th term, $1 + 12(2) = 25$.
Step 3: Add everything up. \[ \text{Sum} = 12 \times 50 + 25 = 600 + 25 = 625 \] \[ \boxed{625} \]