Given the information about the arithmetic sequence, let's solve for the common difference ($d$) and the eleventh term.

Step 1: Understanding the problem

• The third term ($a_3$) is given as $-\frac{2}{3}$.
• The eighth term ($a_8$) is given as $-\frac{22}{3}$.

Step 2: Use the formula for the $n$th term of an arithmetic sequence

The formula for the $n$th term of an arithmetic sequence is: $a_n = a_1 + (n-1)d$ Where:

• $a_n$ is the $n$th term,
• $a_1$ is the first term,
• $d$ is the common difference,
• $n$ is the term number.

Step 3: Set up equations using the known terms

For the third term ($a_3$): $a_3 = a_1 + 2d = -\frac{2}{3}$ For the eighth term ($a_8$): $a_8 = a_1 + 7d = -\frac{22}{3}$

Step 4: Subtract the equations to eliminate $a_1$ and solve for $d$

Subtract the first equation from the second: $(a_1 + 7d) - (a_1 + 2d) = -\frac{22}{3} - \left(-\frac{2}{3}\right)$ Simplifying: $5d = -\frac{20}{3}$ $d = -\frac{20}{3} \div 5 = -\frac{4}{3}$

So, the common difference $d$ is $-\frac{4}{3}$.

Step 5: Find the first term ($a_1$)

Substitute $d = -\frac{4}{3}$ back into the equation for $a_3$: $a_1 + 2\left(-\frac{4}{3}\right) = -\frac{2}{3}$ $a_1 - \frac{8}{3} = -\frac{2}{3}$ $a_1 = -\frac{2}{3} + \frac{8}{3} = \frac{6}{3} = 2$

So, the first term $a_1$ is 2.

Step 6: Find the eleventh term ($a_{11}$)

Using the formula for the $n$th term again: $a_{11} = a_1 + 10d = 2 + 10\left(-\frac{4}{3}\right)$ $a_{11} = 2 - \frac{40}{3} = \frac{6}{3} - \frac{40}{3} = -\frac{34}{3}$

Final Answers:

• The common difference $d$ is $-\frac{4}{3}$.
• The eleventh term $a_{11}$ is $-\frac{34}{3}$.

Would you like more details or have any questions?

Here are 5 related questions for practice:

1. How would you find the first term if only the fourth and ninth terms were known?
2. What would be the sixth term in this sequence?
3. How does changing the common difference affect the sequence?
4. Can you find the sum of the first 20 terms of this arithmetic sequence?
5. What would happen if the common difference was positive instead?

Tip: In arithmetic sequences, the common difference determines whether the sequence is increasing or decreasing.