Problem:
Jerry writes down all the odd numbers 1, 3, 5, 7,..., up to 999. What is the sum of all the numbers he writes down?
Let's break down the solution step-by-step:
1. Identify the sequence:
The sequence of odd numbers from 1 to 999 forms an arithmetic series where each term after the first one is obtained by adding a common difference. In this case:
- The first term ([tex]\(a\)[/tex]) is 1.
- The common difference ([tex]\(d\)[/tex]) is 2.
2. Determine the number of terms in the series:
The last term ([tex]\(l\)[/tex]) is 999.
To find the number of terms ([tex]\(n\)[/tex]), we use the formula for the [tex]\(n\)[/tex]th term of an arithmetic series, which is:
[tex]\[ l = a + (n-1)d \][/tex]
Plugging in the given values,
[tex]\[ 999 = 1 + (n-1) \cdot 2 \][/tex]
Solving for [tex]\(n\)[/tex],
[tex]\[ 999 = 1 + 2n - 2 \][/tex]
[tex]\[ 999 + 1 = 2n \][/tex]
[tex]\[ 1000 = 2n \][/tex]
[tex]\[ n = 500 \][/tex]
So, there are 500 terms in the series.
3. Sum of the arithmetic series:
The sum ([tex]\(S_n\)[/tex]) of the first [tex]\(n\)[/tex] terms of an arithmetic series is given by:
[tex]\[ S_n = \frac{n}{2} \cdot (a + l) \][/tex]
Plugging in the values:
[tex]\[ n = 500 \][/tex]
[tex]\[ a = 1 \][/tex]
[tex]\[ l = 999 \][/tex]
[tex]\[ S_{500} = \frac{500}{2} \cdot (1 + 999) \][/tex]
[tex]\[ S_{500} = 250 \cdot 1000 \][/tex]
[tex]\[ S_{500} = 250000 \][/tex]
Conclusion:
The sum of all the odd numbers from 1 to 999 is 250,000.
