The sum of the first 25 numbers is 325.
We are looking for the sum of the first 25 counting numbers, which means the sum of the integers from 1 to 25. This can be calculated efficiently using the following method outlined in the provided reference:
The method suggests writing the series in both ascending and descending order:
1 + 2 + 3 + ... + 23 + 24 + 25
25 + 24 + 23 + ... + 3 + 2 + 1
Pairing the numbers from both series vertically, we see that each pair sums to 26:
(1+25), (2+24), (3+23), ..., (23+3), (24+2), (25+1).
Since there are 25 such pairs, the sum of the two series is 25 26. As we've counted the original series twice, we must divide the result by 2. Therefore, the sum is (25 26) / 2 = 650 / 2 = 325.
Here’s a more detailed way of looking at the calculation:
- We have 25 pairs that each sum to 26.
- The combined sum of the two series is 25 * 26 = 650
- Since the original series was counted twice, we divide this combined sum by 2.
- Therefore, the sum of numbers from 1 to 25 is 650 / 2 = 325.
| Method | Description |
|---|---|
| Pairing | Pairing numbers from the beginning and end of the sequence and adding them. |
| Calculation | Multiply the sum of the first and last number (1+25 = 26) by half of the number of terms (25/2 = 12.5) which gives 26 * 12.5=325. |
