We are given an arithmetic sequence: $22, 11, 0, -11, \dots, -77$. To find the sum, let's follow these steps:

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

The common difference is the difference between any two consecutive terms: $d = 11 - 22 = -11$

Step 2: Identify the first term ($a$) and the last term ($l$)

• The first term is $a = 22$.
• The last term given is $l = -77$.

Step 3: Find the number of terms ($n$)

The $n$-th term of an arithmetic sequence is given by: $a_n = a + (n - 1) \cdot d$ We know $a_n = -77$, so let's plug in the values: $-77 = 22 + (n - 1) \cdot (-11)$ Simplify and solve for $n$: $-77 = 22 - 11(n - 1)$ $-77 = 22 - 11n + 11$ $-77 = 33 - 11n$ $-77 - 33 = -11n$ $-110 = -11n$ $n = 10$ So, there are 10 terms in the sequence.

Step 4: Use the sum formula for an arithmetic sequence

The sum $S_n$ of an arithmetic sequence is given by: $S_n = \frac{n}{2} \cdot (a + l)$ Substitute the known values: $S_{10} = \frac{10}{2} \cdot (22 + (-77))$ $S_{10} = 5 \cdot (-55) = -275$

Final Answer:

The sum of the arithmetic sequence is $-275$.

Would you like more details on any of the steps?

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Here are 5 related questions for further practice:

1. What is the sum of the first 15 terms of the arithmetic sequence $7, 4, 1, \dots$?
2. If the common difference in an arithmetic sequence is 5 and the first term is 10, what is the 12th term?
3. How many terms are in the arithmetic sequence $50, 45, 40, \dots, -30$?
4. Find the sum of the arithmetic sequence $2, 7, 12, \dots, 97$.
5. What is the sum of the first 20 terms of the arithmetic sequence $100, 90, 80, \dots$?

Tip: When dealing with arithmetic sequences, always identify the first term, common difference, and number of terms first to simplify your calculations.