To find the sum of the first 10 terms of the arithmetic sequence, we need to follow these steps:

Step 1: Determine the common difference ($d$)

The general formula for the $n$th term of an arithmetic sequence is:

$a_n = a_1 + (n - 1) \cdot d$

Given:

• The 3rd term ($a_3$) is $-12$.
• The 7th term ($a_7$) is $8$.

Using the formula for the 3rd term:

$a_3 = a_1 + 2d = -12 \quad \text{(1)}$

Using the formula for the 7th term:

$a_7 = a_1 + 6d = 8 \quad \text{(2)}$

Step 2: Solve the system of equations to find $a_1$ and $d$

Subtract equation (1) from equation (2):

$(a_1 + 6d) - (a_1 + 2d) = 8 - (-12)$

$4d = 20$

$d = 5$

Now, substitute $d = 5$ back into equation (1):

$a_1 + 2(5) = -12$

$a_1 + 10 = -12$

$a_1 = -22$

So, the first term $a_1$ is $-22$ and the common difference $d$ is $5$.

Step 3: Find the sum of the first 10 terms

The sum $S_n$ of the first $n$ terms of an arithmetic sequence is given by:

$S_n = \frac{n}{2} \cdot (2a_1 + (n-1) \cdot d)$

Substitute $n = 10$, $a_1 = -22$, and $d = 5$:

$S_{10} = \frac{10}{2} \cdot (2(-22) + 9 \cdot 5)$

$S_{10} = 5 \cdot (-44 + 45)$

$S_{10} = 5 \cdot 1 = 5$

Final Answer:

The sum of the first 10 terms of the arithmetic sequence is 5.

Would you like further details or have any questions?

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Here are 5 related questions you might find interesting:

1. How would the sum change if the common difference was doubled?
2. What is the 10th term of the sequence?
3. Can you determine the sum of the first 15 terms with the same common difference?
4. How do you find the sum of an infinite arithmetic series?
5. What happens to the sum if all the terms are multiplied by a constant?

Tip: When dealing with arithmetic sequences, identifying the common difference is crucial, as it directly influences both individual terms and their sums.