What is the sum of 25 consecutive odd numbers?

The sum of 25 consecutive odd numbers is 625.

The provided reference states that there are 25 odd numbers between 1 and 50. Furthermore, it explicitly mentions that the sum of the first n odd numbers is n². Therefore, the sum of the first 25 odd numbers (1, 3, 5, ..., 49) is 25² = 625.

Here is a breakdown of how this is derived:

• Formula: The sum of the first n odd numbers is given by the formula S_n = n².
• Application: In our case, n = 25.
• Calculation: Therefore, S₂₅ = 25² = 625.

The reference also indicates an alternative method, which involves using the formula for the sum of an arithmetic series: S_n = n/2 × (a + l), where a is the first term and l is the last term.

Let's verify with that alternative method:

1. The first odd number is 1, so a = 1.
2. The 25th odd number is 49, so l = 49.
3. Applying the formula for 25 odd numbers: S₂₅ = 25/2 × (1 + 49)
4. S₂₅ = 25/2 × 50
5. S₂₅ = 25 × 25
6. S₂₅ = 625

Both methods lead to the same answer, confirming that the sum of the first 25 consecutive odd numbers is indeed 625.