Problem 17: Given an arithmetic sequence $\{a_n\}$ with $a_5 = 1$ and $a_8 = 8$, find the common difference $q$. Problem 18: Given a geometric sequence $\{a_n\}$ where $S_n$ represents the sum of the first $n$ terms, and $S_9 = 46$, find $a_5$.

1. Problem Description

Problem 17: Given an arithmetic sequence $\{a_n\}$ with $a_5 = 1$ and $a_8 = 8$, find the common difference $q$.

Problem 18: Given a geometric sequence $\{a_n\}$ where $S_n$ represents the sum of the first $n$ terms, and $S_9 = 46$, find $a_5$.

2. Solution Steps

Problem 17:

Since $\{a_n\}$ is an arithmetic sequence, we can express $a_n$ as:

$a_n = a_1 + (n-1)d$, where $d$ is the common difference.

Given $a_5 = 1$ and $a_8 = 8$, we have:

$a_5 = a_1 + 4d = 1$

$a_8 = a_1 + 7d = 8$

Subtracting the first equation from the second equation, we get:

$(a_1 + 7d) - (a_1 + 4d) = 8 - 1$

$3d = 7$

$d = \frac{7}{3}$

The problem asks for the common ratio q, but this sequence is given to be an arithmetic progression, thus there can't be a common ratio. Since we found the common difference d, there must be a typo and it meant to ask for the common difference instead.

Problem 18:

Given that $S_9 = 46$, we want to find $a_5$.

For a geometric sequence, we can rewrite $a_5$ as $a_1q^4$. However, we only have $S_9$. It looks like something is missing in the prompt. It is impossible to find $a_5$ with this information. Without more details on the question such as $a_1$, or $q$, it cannot be determined what $a_5$ is.

However, let's assume the question meant $S_5=46$ and wanted to know $a_5$, given that it is a geometric sequence. That would be impossible as well.

3. Final Answer

Problem 17: $\frac{7}{3}$

Problem 18: Cannot be determined with the given information.