Question

The sum of 100 consecutive positive whole numbers is 5,350. What is the first number?

Answer 1

Let the first number of the 100 consecutive positive whole numbers be \( n \). The consecutive numbers can then be represented as \( n, n+1, n+2, \ldots, n+99 \).

To find the sum of these 100 numbers, we use the formula for the sum of an arithmetic series. The sum \( S \) of \( k \) consecutive integers starting at \( a \) is given by:

\[
S = \frac{k}{2} \times (\text{first term} + \text{last term})
\]

In this case, \( k = 100 \), the first term is \( n \), and the last term is \( n + 99 \). Therefore, the sum can be calculated as follows:

\[
S = \frac{100}{2} \times (n + (n + 99))
\]
\[
S = 50 \times (2n + 99)
\]
\[
S = 100n + 4950
\]

Given that the sum \( S \) is 5350, we set up the equation:

\[
100n + 4950 = 5350
\]

Next, we solve for \( n \):

\[
100n = 5350 - 4950
\]
\[
100n = 400
\]
\[
n = \frac{400}{100} = 4
\]

Thus, the first number is

\[
\boxed{4}
\]

To verify, we can calculate the sum of the 100 consecutive numbers starting from 4:

The numbers are \( 4, 5, 6, \ldots, 103 \), and sum can be calculated as follows:

\[
\text{Last term} = 4 + 99 = 103
\]

Calculating the sum:

\[
S = \frac{100}{2} \times (4 + 103) = 50 \times 107 = 5350
\]

The calculation confirms that the first number is indeed

\[
\boxed{4}
\]