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Let's break down and solve the given problem step by step. 

We are given a recursive sequence \( \{a_n\} \) defined as:

\[
a_1 = 2
\]
\[
a_{n+1} = 2(a_1 + a_2 + \cdots + a_n) \quad \text{for} \quad n = 1, 2, 3, \dots
\]

### (i) Evaluate \( a_2 \), \( a_3 \), and \( a_4 \).

We are provided the initial term \( a_1 = 2 \).

To find the next terms:

- \( a_2 = 2(a_1) = 2(2) = 4 \)
- \( a_3 = 2(a_1 + a_2) = 2(2 + 4) = 2(6) = 12 \)
- \( a_4 = 2(a_1 + a_2 + a_3) = 2(2 + 4 + 12) = 2(18) = 36 \)

Thus, the values are:
\[
a_2 = 4, \quad a_3 = 12, \quad a_4 = 36
\]

### (ii) Find the ratio \( \frac{a_{n+1}}{a_n} \) where \( n \geq 2 \). Is \( \{a_n\}_{n=2}^{\infty} \) a geometric sequence?

We will calculate the ratios for the given terms:
\[
\frac{a_2}{a_1} = \frac{4}{2} = 2
\]
\[
\frac{a_3}{a_2} = \frac{12}{4} = 3
\]
\[
\frac{a_4}{a_3} = \frac{36}{12} = 3
\]

Notice that starting from \( n = 3 \), the ratio becomes constant at 3. This implies that for \( n \geq 3 \), the sequence behaves as a geometric sequence with a common ratio of 3.

Therefore, \( \{a_n\}_{n=2}^{\infty} \) is a geometric sequence starting from \( n = 3 \) with the ratio \( r = 3 \).

### (iii) Find the sum of the first \( n \) terms \( a_1 + a_2 + \cdots + a_n \).

We will use the fact that \( a_{n+1} = 2(a_1 + a_2 + \cdots + a_n) \) to find a pattern for the sum.

Let \( S_n = a_1 + a_2 + \cdots + a_n \).

From the recursive formula:
\[
a_2 = 2a_1 \quad \Rightarrow \quad S_2 = a_1 + a_2 = 2 + 4 = 6
\]
\[
a_3 = 2(a_1 + a_2) = 2S_2 \quad \Rightarrow \quad S_3 = a_1 + a_2 + a_3 = S_2 + a_3 = 6 + 12 = 18
\]
\[
a_4 = 2(a_1 + a_2 + a_3) = 2S_3 \quad \Rightarrow \quad S_4 = a_1 + a_2 + a_3 + a_4 = S_3 + a_4 = 18 + 36 = 54
\]

The sum follows the pattern \( S_n = 3S_{n-1} \), making it a growing sum that triples at each stage.

---

Would you like any additional clarification on any part of the solution?

Here are five related questions to explore further:
1. How does the recursive formula affect the growth rate of the sequence?
2. Can you generalize the formula for \( S_n \) based on the pattern we observed?
3. What conditions are necessary for a sequence to be classified as geometric?
4. How would the solution change if the initial condition \( a_1 \) were different?
5. Can you express the sum \( S_n \) explicitly without recursion?

**Tip:** When working with recursive sequences, always calculate a few initial terms to detect any patterns or behaviors that might simplify the solution.

The sequence {a_n} is defined as follows: a_1 = 2, a_{n+1} = 2(a_1 + a_2 + ... + a_n) for n = 1, 2, 3,... 
(i) Evaluate the numerical value of a_2, a_3 and a_4.
(ii) Find the ratio of a_{n+1} / a_n where n >= 2. Is {a_n} for n >= 2 a geometric sequence?
(iii) Find the sum of the n terms a_1 + a_2 + ... + a_n.

Explore how to solve a recursive sequence problem with a step-by-step solution. Evaluate specific terms, determine if the sequence forms a geometric progression, and find the sum of the first n terms.

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