To solve the problem of how many odd numbers Jerry writes down when listing all odd numbers from 1 to 999, we can follow a methodical approach using the concept of arithmetic sequences.
1. Identify the properties of the sequence:
- The sequence starts at 1.
- The common difference between consecutive terms is 2 (since each number differs from the previous one by 2).
2. Formulate the general form of the nth term of the sequence:
- The nth term ([tex]\(a_n\)[/tex]) of an arithmetic sequence can be described by the formula:
[tex]\[
a_n = a_1 + (n - 1) \cdot d
\][/tex]
Where [tex]\(a_1\)[/tex] is the first term, [tex]\(d\)[/tex] is the common difference, and [tex]\(n\)[/tex] is the number of terms.
3. Assign the known values:
- The first term ([tex]\(a_1\)[/tex]) is 1.
- The common difference ([tex]\(d\)[/tex]) is 2.
- The last term ([tex]\(a_n\)[/tex]) is 999.
4. Set up the equation to find the number of terms (n):
[tex]\[
999 = 1 + (n - 1) \cdot 2
\][/tex]
5. Solve for [tex]\(n\)[/tex]:
- First, isolate the term involving [tex]\(n\)[/tex]:
[tex]\[
999 = 1 + 2(n - 1)
\][/tex]
[tex]\[
999 - 1 = 2(n - 1)
\][/tex]
[tex]\[
998 = 2(n - 1)
\][/tex]
- Next, divide both sides by 2:
[tex]\[
\frac{998}{2} = n - 1
\][/tex]
[tex]\[
499 = n - 1
\][/tex]
- Finally, solve for [tex]\(n\)[/tex] by adding 1 to both sides:
[tex]\[
n = 499 + 1
\][/tex]
[tex]\[
n = 500
\][/tex]
Therefore, Jerry writes down 500 odd numbers in the sequence from 1 to 999.
