To find the sum of the first 36 terms of an arithmetic sequence, we use the formula for the sum of an arithmetic sequence:

$S_n = \frac{n}{2} \times (a + l)$

Where:

• $S_n$ is the sum of the first $n$ terms,
• $n$ is the number of terms,
• $a$ is the first term,
• $l$ is the last term.

Step 1: Identify the given values

From the problem:

• The first term $a = 22$,
• The common difference $d$ can be calculated: $d = 27 - 22 = 5.$
• The number of terms $n = 36$.

To find the last term $l$, use the formula for the $n$-th term of an arithmetic sequence:

$l = a + (n-1) \cdot d$

Step 2: Calculate the last term ($l$)

$l = 22 + (36-1) \cdot 5$ $l = 22 + 35 \cdot 5$ $l = 22 + 175 = 197$

The last term $l = 197$.

Step 3: Find the sum of the first 36 terms ($S_{36}$)

Now substitute into the sum formula:

$S_n = \frac{n}{2} \cdot (a + l)$ $S_{36} = \frac{36}{2} \cdot (22 + 197)$ $S_{36} = 18 \cdot 219$ $S_{36} = 3942$

Final Answer:

The sum of the first 36 terms is 3942.

Do you need further clarification or a breakdown of any step? Let me know!

Related Questions:

1. How do you derive the formula for the sum of an arithmetic sequence?
2. How can you find the $n$-th term of an arithmetic sequence?
3. If the common difference changes, how would the sum be affected?
4. What if the sequence had a different number of terms, such as 50 terms?
5. How can we determine whether a sequence is arithmetic or not?

Tip: Always verify the last term calculation when working with arithmetic sequences to avoid small errors.