The question in the image asks for the sum of the first 10 terms of an arithmetic sequence, where three of the terms are given: 8, $x$, and 22. Let's solve this step by step:

1. Identify the common difference

In an arithmetic sequence, the difference between consecutive terms is constant. Let's denote this common difference as $d$.

The terms given are:

• First term $a_1 = 8$
• Third term $a_3 = 22$

We can use the formula for the $n$-th term of an arithmetic sequence: $a_n = a_1 + (n - 1) \cdot d$

For the third term: $a_3 = a_1 + 2d$

Substituting the values: $22 = 8 + 2d$ $2d = 22 - 8 = 14$ $d = 7$

2. Calculate the second term

Now that we know the common difference is 7, we can find the second term $a_2$: $a_2 = a_1 + d = 8 + 7 = 15$

So, the sequence is: $8, 15, 22, \dots$

3. Sum of the first 10 terms

The formula for the sum of the first $n$ terms of an arithmetic sequence is: $S_n = \frac{n}{2} \cdot (a_1 + a_n)$

For $n = 10$, we need the 10th term $a_{10}$: $a_{10} = a_1 + (10 - 1) \cdot d = 8 + 9 \cdot 7 = 8 + 63 = 71$

Now, calculate the sum: $S_{10} = \frac{10}{2} \cdot (8 + 71) = 5 \cdot 79 = 395$

Final Answer: The sum of the first 10 terms is 395, so the correct choice is A.

Would you like more details on any part of the solution?

Here are some related questions to explore further:

1. How do you find the sum of the first $n$ terms of any arithmetic sequence?
2. What happens if the common difference is negative?
3. Can an arithmetic sequence have non-integer terms?
4. What is the formula for the sum of terms in a geometric sequence?
5. How would you adjust the formula for finding the sum if the sequence starts with a negative number?

Tip: Always check that the terms in an arithmetic sequence maintain a constant difference!