What will be the sum of odd numbers from 1 to 50?

Question

Accepted Answer

To determine the sum of odd numbers from 1 to 50, we need to first identify the sequence of odd numbers within this range. The odd numbers from 1 to 50 are: 1, 3, 5, 7, ..., 49.

This sequence is an arithmetic series where the first term $ a = 1 $, the last term $ l = 49 $, and the common difference $ d = 2 $.

The formula for the sum $ S_n $ of an arithmetic series is:

$$ S_n = \frac{n}{2} \times (a + l) $$

First, we need to find the number of terms $ n $ in the series. The formula for the $ n $-th term of an arithmetic sequence is:

$$ a_n = a + (n-1) \times d $$

Setting $ a_n = 49 $, we solve for $ n $:

$$ 49 = 1 + (n-1) \times 2 $$

$$ 49 = 1 + 2n - 2 $$

$$ 49 = 2n - 1 $$

$$ 50 = 2n $$

$$ n = 25 $$

Now, we can calculate the sum using the sum formula:

$$ S_{25} = \frac{25}{2} \times (1 + 49) $$

$$ S_{25} = \frac{25}{2} \times 50 $$

$$ S_{25} = 25 \times 25 $$

$$ S_{25} = 625 $$

Thus, the sum of odd numbers from 1 to 50 is 625, which confirms that Option 1 is correct.

**Explanation for Incorrect Options:**

- **Option 2 (650):** This option is incorrect because the calculation of the sum of the series using the arithmetic series formula results in 625, not 650. The arithmetic progression and the number of terms have been correctly calculated, and 650 does not match the result of the formula.

- **Option 3 (667):** This option is incorrect for the same reason as Option 2. The sum of the series calculated using the correct arithmetic progression formula is 625, and 667 is not the result of this calculation.

- **Option 4 (670):** This option is incorrect because, as with the other incorrect options, the sum of the odd numbers from 1 to 50 is 625, not 670. The arithmetic series formula has been applied correctly, and 670 does not match the calculated sum.

In conclusion, the correct answer is Option 1, as the sum of the odd numbers from 1 to 50 is indeed 625.