Sure, let's solve for the first three-digit even number in each sequence.
### Sequence [tex]\(a_n = \frac{n^2}{4}\)[/tex]
We need to find the smallest [tex]\(n\)[/tex] such that [tex]\(\frac{n^2}{4}\)[/tex] is a three-digit number and even.
1. First, understand the range of three-digit numbers: Between 100 and 999.
2. Since we need [tex]\(\frac{n^2}{4}\)[/tex] to be even, let's set up the inequality:
[tex]\[
100 \leq \frac{n^2}{4} < 1000
\][/tex]
3. Multiply through by 4:
[tex]\[
400 \leq n^2 < 4000
\][/tex]
4. Take the square root of the entire inequality:
[tex]\[
20 \leq n < 63.245
\][/tex]
Next, we need to check the values in this range to see which [tex]\(n\)[/tex] gives even results when [tex]\(\frac{n^2}{4}\)[/tex] is calculated.
Calculate [tex]\(\frac{n^2}{4}\)[/tex] for values starting from 20 and check if it is a three-digit even number.
- For [tex]\(n = 20\)[/tex]:
[tex]\[
a_{20} = \frac{20^2}{4} = \frac{400}{4} = 100 \quad \text{which is even and three-digit}.
\][/tex]
Thus, the first three-digit even number in this sequence is [tex]\(100\)[/tex].
### Sequence [tex]\(a_n = \frac{n^3}{2}\)[/tex]
We need to find the smallest [tex]\(n\)[/tex] such that [tex]\(\frac{n^3}{2}\)[/tex] is a three-digit number and even.
1. Begin with the range of three-digit numbers: Between 100 and 999.
2. Since we need [tex]\(\frac{n^3}{2}\)[/tex] to be even, set up the inequality:
[tex]\[
100 \leq \frac{n^3}{2} < 1000
\][/tex]
3. Multiply through by 2:
[tex]\[
200 \leq n^3 < 2000
\][/tex]
4. Take the cube root of the entire inequality:
[tex]\[
5.84 \leq n < 12.599
\][/tex]
Next, we need to check the values in this range to see which [tex]\(n\)[/tex] gives even results when [tex]\(\frac{n^3}{2}\)[/tex] is calculated.
Calculate [tex]\(\frac{n^3}{2}\)[/tex] for values starting from 6 (since [tex]\(n\)[/tex] should be an integer starting from 6) and check if it is a three-digit even number.
- For [tex]\(n = 6\)[/tex]:
[tex]\[
a_{6} = \frac{6^3}{2} = \frac{216}{2} = 108 \quad \text{which is even and three-digit}.
\][/tex]
Thus, the first three-digit even number in this sequence is [tex]\(108\)[/tex].
### Conclusion
- For sequence [tex]\(a_n = \frac{n^2}{4}\)[/tex], the first three-digit even number is [tex]\(100\)[/tex].
- For sequence [tex]\(a_n = \frac{n^3}{2}\)[/tex], the first three-digit even number is [tex]\(108\)[/tex].
