Let's break down the problem step by step.

Given:

• The third term of the arithmetic sequence ($a_3$) is $-8$.
• The sum of the first 10 terms ($S_{10}$) is $-230$.

To find:

a) The first term of the sequence ($a_1$)
b) The sum of the first 13 terms ($S_{13}$)

Step 1: Use the formula for the $n$th term of an arithmetic sequence

The general formula for the $n$th term of an arithmetic sequence is: $a_n = a_1 + (n-1)d$ For the third term ($n = 3$): $a_3 = a_1 + 2d$ Given $a_3 = -8$, we have: $-8 = a_1 + 2d \quad \text{(Equation 1)}$

Step 2: Use the formula for the sum of the first $n$ terms

The formula for the sum of the first $n$ terms is: $S_n = \frac{n}{2} \times (2a_1 + (n-1)d)$ For the first 10 terms ($n = 10$): $S_{10} = \frac{10}{2} \times (2a_1 + 9d) = 5 \times (2a_1 + 9d)$ Given $S_{10} = -230$, we have: $5 \times (2a_1 + 9d) = -230$ Dividing both sides by 5: $2a_1 + 9d = -46 \quad \text{(Equation 2)}$

Step 3: Solve the system of equations

We have two equations:

1. $a_1 + 2d = -8$
2. $2a_1 + 9d = -46$

We can solve this system of equations to find $a_1$ and $d$.

First, solve Equation 1 for $a_1$: $a_1 = -8 - 2d$ Substitute this into Equation 2: $2(-8 - 2d) + 9d = -46$ Expand and simplify: $-16 - 4d + 9d = -46$ $5d = -30$ $d = -6$

Now, substitute $d = -6$ back into the expression for $a_1$: $a_1 = -8 - 2(-6) = -8 + 12 = 4$

So, the first term ($a_1$) is 4.

Step 4: Find the sum of the first 13 terms

Use the sum formula $S_{13}$: $S_{13} = \frac{13}{2} \times (2a_1 + 12d)$ Substitute the values of $a_1 = 4$ and $d = -6$: $S_{13} = \frac{13}{2} \times (2(4) + 12(-6)) = \frac{13}{2} \times (8 - 72) = \frac{13}{2} \times (-64) = 13 \times (-32) = -416$

Final Answers:

a) The first term of the sequence ($a_1$) is 4.
b) The sum of the first 13 terms ($S_{13}$) is -416.

Would you like any further explanation or details?

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Relative Questions:

1. How is the common difference calculated in an arithmetic sequence?
2. What is the formula for the sum of the first $n$ terms of an arithmetic sequence?
3. How does the value of the common difference affect the sequence?
4. Can you find the 20th term of this sequence?
5. How would the problem change if the sum of the first 10 terms were a different value?

Tip: When solving systems of linear equations, always check your solutions by substituting them back into the original equations.