We are given an arithmetic sequence: 22, 11, 0, −11, …, −77, and asked to find the sum of the terms.

Step 1: Identify the necessary parameters

1. First term (a): The first term is 22.
2. Common difference (d): The difference between consecutive terms is: $d = 11 - 22 = -11$
3. Last term (l): The last term given is −77.

Step 2: Find the number of terms (n)

The formula for the $n$-th term of an arithmetic sequence is: $a_n = a + (n - 1) \cdot d$ We know that $a_n = -77$, $a = 22$, and $d = -11$. Substituting these values into the formula: $-77 = 22 + (n - 1) \cdot (-11)$ Simplifying: $-77 = 22 - 11(n - 1)$ $-77 - 22 = -11(n - 1)$ $-99 = -11(n - 1)$ Dividing both sides by -11: $n - 1 = 9$ $n = 10$ So, there are 10 terms in the sequence.

Step 3: Use the sum formula for an arithmetic sequence

The sum of the first $n$ terms of an arithmetic sequence is given by the formula: $S_n = \frac{n}{2} \cdot (a + l)$ Substituting $n = 10$, $a = 22$, and $l = -77$: $S_{10} = \frac{10}{2} \cdot (22 + (-77)) = 5 \cdot (22 - 77) = 5 \cdot (-55) = -275$

Thus, the sum of the sequence is $\boxed{-275}$.

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Do you want more details or have any other questions?

Here are some related questions for practice:

1. Find the sum of the arithmetic sequence with first term 15, common difference -5, and last term -25.
2. What is the sum of the first 20 terms of the arithmetic sequence where $a = 7$ and $d = 3$?
3. How many terms are there in the arithmetic sequence 10, 7, 4, ..., -50?
4. Find the sum of the arithmetic sequence: 3, 8, 13, ..., 98.
5. Determine the common difference of an arithmetic sequence if the first term is 6 and the 10th term is -18.

Tip: Always check the number of terms in an arithmetic sequence before applying the sum formula!