To determine if 88 will be in the sequence 8, 16, 24, 32, 40..., we need to identify whether 88 can be written in the form of an arithmetic sequence with the given terms. This sequence appears to follow an arithmetic pattern where each term increases by a constant difference.
1. Identify the first term and common difference:
- The first term [tex]\(a\)[/tex] of the sequence is 8.
- The common difference [tex]\(d\)[/tex] between consecutive terms is 16 - 8 = 8.
2. General form of an arithmetic sequence:
- The nth term ([tex]\(a_n\)[/tex]) of an arithmetic sequence can be expressed using the formula:
[tex]\[
a_n = a + (n-1) \cdot d
\][/tex]
- For this sequence, it becomes:
[tex]\[
a_n = 8 + (n-1) \cdot 8
\][/tex]
3. Set up the equation to solve for [tex]\(n\)[/tex]:
- We want to check if 88 is one of the terms in the sequence. Therefore, we set [tex]\(a_n = 88\)[/tex]:
[tex]\[
88 = 8 + (n-1) \cdot 8
\][/tex]
4. Solve for [tex]\(n\)[/tex]:
- Subtract 8 from both sides:
[tex]\[
88 - 8 = (n-1) \cdot 8
\][/tex]
[tex]\[
80 = (n-1) \cdot 8
\][/tex]
- Divide both sides by 8:
[tex]\[
\frac{80}{8} = n-1
\][/tex]
[tex]\[
10 = n-1
\][/tex]
- Add 1 to both sides:
[tex]\[
n = 11
\][/tex]
5. Conclusion:
- The value [tex]\(n = 11\)[/tex] indicates that 88 is the 11th term in the sequence. Since [tex]\(n = 11\)[/tex] is a positive integer, this confirms that 88 is indeed in the sequence.
Therefore, 88 will be in the sequence. It specifically appears as the 11th term in this arithmetic progression.
