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: Understanding the Question:
This question is from Arithmetic Progressions (AP), specifically focusing on calculating the sum of a sequence of odd integers.

Step 2: Key Formula or Approach:
The sum of the first \(n\) odd natural numbers is given by the standard formula:
\[ S_n = n^2 \]
Alternatively, we can use the general AP sum formula:
\[ S_n = \frac{n}{2} [a + l] \]
where \(a\) is the first term, \(l\) is the last term, and \(n\) is the number of terms.

Step 3: Detailed Explanation:
The odd numbers between 0 and 50 form the following arithmetic progression:
\[ 1, 3, 5, 7, \dots, 49 \]
Here, the first term is:
\[ a = 1 \]
The common difference is:
\[ d = 2 \]
The last term is:
\[ l = 49 \]
Let us find the number of terms \(n\) using the formula for the \(n\)-th term of an AP:
\[ a_n = a + (n-1)d \]
Substitute the values:
\[ 49 = 1 + (n-1)2 \]
\[ 48 = 2(n-1) \]
\[ n-1 = 24 \implies n = 25 \]
Now, compute the sum of these 25 terms:
Using the formula \(S_n = n^2\):
\[ S_{25} = 25^2 = 625 \]
Let us verify using the other sum formula:
\[ S_{25} = \frac{25}{2} [1 + 49] \]
\[ S_{25} = \frac{25}{2} [50] = 25 \times 25 = 625 \]
Both methods yield the same result.

Step 4: Final Answer:
The sum of the odd numbers between 0 and 50 is \(625\).