This problem involves a recursive sequence $u_n$ where:

• $u_1 = 10$
• $u_{n+1} = u_n + 10$

(a) Find the first three terms:

The recursive rule tells us how to find each successive term by adding 10 to the previous term. Starting from $u_1 = 10$:

• $u_1 = 10$
• $u_2 = u_1 + 10 = 10 + 10 = 20$
• $u_3 = u_2 + 10 = 20 + 10 = 30$

Thus, the first three terms are: $u_1 = 10, \ u_2 = 20, \ u_3 = 30$

(b) Find $S_1$, $S_2$, and $S_3$:

$S_n$ represents the sum of the first $n$ terms of the sequence $u_n$. The sum of the first $n$ terms is denoted as: $S_n = u_1 + u_2 + \cdots + u_n$

Now, calculate the required sums:

• $S_1 = u_1 = 10$
• $S_2 = u_1 + u_2 = 10 + 20 = 30$
• $S_3 = u_1 + u_2 + u_3 = 10 + 20 + 30 = 60$

Thus, the values are: $S_1 = 10, \ S_2 = 30, \ S_3 = 60$

Would you like more details on recursive sequences or need further clarifications? Here are some related questions to consider:

1. How do you find the general form of a recursive arithmetic sequence?
2. What is the formula for the sum of an arithmetic series?
3. How does the difference between terms help in identifying the type of sequence?
4. Can you express $S_n$ as a general formula for any $n$?
5. What would happen if the recursive relation included a multiplicative factor instead of an additive one?

Tip: Recursive sequences are defined step by step, but finding a pattern can often lead to discovering a direct (explicit) formula.