The question from the image is about an arithmetic sequence. Let's break it down step by step:

Given:

• The third term ($U_3$) of an arithmetic sequence is 36.
• The sum of the fifth ($U_5$) and seventh ($U_7$) terms is 144.
• We are asked to find the sum of the first ten terms of the sequence.

Step 1: General formula for the $n$-th term in an arithmetic sequence

The $n$-th term of an arithmetic sequence is given by the formula: $U_n = a + (n-1)d$ where:

• $a$ is the first term,
• $d$ is the common difference,
• $n$ is the position of the term.

Step 2: Using the given information to form equations

1. For the third term ($U_3$): $U_3 = a + 2d = 36$
2. The sum of the fifth and seventh terms: $U_5 + U_7 = (a + 4d) + (a + 6d) = 2a + 10d = 144$

Step 3: Solve the system of equations

From $a + 2d = 36$ and $2a + 10d = 144$, we can solve for $a$ and $d$.

1. $a + 2d = 36$ $\Rightarrow$ $a = 36 - 2d$
2. Substitute $a = 36 - 2d$ into the second equation: $2(36 - 2d) + 10d = 144$ $72 - 4d + 10d = 144$ $72 + 6d = 144$ $6d = 144 - 72 = 72$ $d = 12$

Step 4: Find $a$

Substitute $d = 12$ back into $a + 2d = 36$: $a + 2(12) = 36$ $a + 24 = 36$ $a = 12$

Step 5: Sum of the first 10 terms

The sum $S_n$ of the first $n$ terms of an arithmetic sequence is given by the formula: $S_n = \frac{n}{2} \times (2a + (n-1)d)$ For $n = 10$: $S_{10} = \frac{10}{2} \times (2(12) + 9(12))$ $S_{10} = 5 \times (24 + 108) = 5 \times 132 = 660$

Final Answer:

The sum of the first ten terms is 660. Thus, the correct choice is (D) 660.

Would you like further details on this solution or another explanation? Here are some follow-up questions that you might find helpful:

1. How do we derive the formula for the sum of an arithmetic series?
2. Why do we substitute the third and fifth/seventh terms to create two equations?
3. What are the conditions required for a sequence to be arithmetic?
4. How would the solution change if the common difference were negative?
5. What other methods can be used to find the sum of an arithmetic sequence?

Tip: When solving arithmetic sequence problems, always start by finding expressions for the terms you're given, then solve the system of equations. This will help you determine both the first term and the common difference.