To insert $x$ and $y$ between 36 and 27 to form an arithmetic sequence, we need to find the common difference $d$ and determine $x$ and $y$ accordingly.

Given: $a_1 = 36$ $a_4 = 27$

The general formula for the $n$-th term of an arithmetic sequence is: $a_n = a_1 + (n-1)d$

For four terms, the sequence will be: $a_1 = 36$ $a_2 = x$ $a_3 = y$ $a_4 = 27$

Using the general formula for the fourth term: $a_4 = a_1 + 3d$ $27 = 36 + 3d$

Solving for $d$: $27 - 36 = 3d$ $-9 = 3d$ $d = -3$

Now that we have the common difference $d = -3$, we can find $x$ and $y$.

The second term $a_2 = x$: $a_2 = a_1 + d$ $x = 36 + (-3)$ $x = 33$

The third term $a_3 = y$: $a_3 = a_1 + 2d$ $y = 36 + 2(-3)$ $y = 36 - 6$ $y = 30$

Thus, the sequence is $36, 33, 30, 27$.

To find $x + y$: $x + y = 33 + 30 = 63$

Therefore, $x + y = 63$.

Would you like any further details or have any questions about this solution?

Here are 8 related questions to further expand on the topic:

1. How would you find the common difference if only the first and last terms of an arithmetic sequence are given, along with the number of terms?
2. Can you determine the sum of the four terms in the sequence $36, 33, 30, 27$?
3. How do you find the number of terms in an arithmetic sequence if the first term, last term, and common difference are known?
4. How can you insert more than two numbers to form a longer arithmetic sequence between 36 and 27?
5. What is the general term $a_n$ for the sequence $36, 33, 30, 27$?
6. How do arithmetic sequences relate to linear functions?
7. What are the properties of arithmetic sequences that distinguish them from geometric sequences?
8. How can you determine the missing terms in an arithmetic sequence if some terms are missing and the sequence is partially known?

Tip: When working with sequences, always verify your results by plugging the terms back into the sequence formula to ensure consistency.