The sequence {an} is defined as follows:
a1 = 2, an+1 = 2(a1 + a2 + ⋅ ⋅ ⋅ + an) for n = 1, 2, 3 . . .
(i) [5 marks] Evaluate the numerical value of a2, a3 and a4.
(ii) [5 marks] Find the ratio of an+1
an
where n ≥ 2. Is {an}
∞
n=2
a
geometric sequence?

We are given the sequence \(\{a_n\}\) defined as follows:

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

### Part (i): Evaluate the numerical values of \(a_2\), \(a_3\), and \(a_4\)

We will compute the first few terms of the sequence.

- \(a_1 = 2\)
- To find \(a_2\), we use the formula:
  \[
  a_2 = 2(a_1) = 2(2) = 4
  \]
- To find \(a_3\), we use:
  \[
  a_3 = 2(a_1 + a_2) = 2(2 + 4) = 2(6) = 12
  \]
- To find \(a_4\), we use:
  \[
  a_4 = 2(a_1 + a_2 + a_3) = 2(2 + 4 + 12) = 2(18) = 36
  \]

Thus, the first four terms of the sequence are:
\[
a_1 = 2, \quad a_2 = 4, \quad a_3 = 12, \quad a_4 = 36
\]

### Part (ii): Find the ratio \(\frac{a_{n+1}}{a_n}\) and determine if the sequence is geometric

To find the ratio \(\frac{a_{n+1}}{a_n}\) for \(n \geq 2\), we calculate the ratios for specific values of \(n\).

- For \(n = 2\):
  \[
  \frac{a_3}{a_2} = \frac{12}{4} = 3
  \]
- For \(n = 3\):
  \[
  \frac{a_4}{a_3} = \frac{36}{12} = 3
  \]

In both cases, the ratio \(\frac{a_{n+1}}{a_n} = 3\). This suggests that the ratio is constant for \(n \geq 2\).

#### Is \(\{a_n\}\) a geometric sequence?

A sequence is geometric if the ratio between consecutive terms is constant. Since we have found that the ratio \(\frac{a_{n+1}}{a_n} = 3\) for \(n \geq 2\), the sequence \(\{a_n\}\) for \(n \geq 2\) is indeed a **geometric sequence** with common ratio 3.

### Summary:
- (i) The numerical values are:
  \[
  a_2 = 4, \quad a_3 = 12, \quad a_4 = 36
  \]
- (ii) The ratio \(\frac{a_{n+1}}{a_n} = 3\) for \(n \geq 2\), and the sequence \(\{a_n\}\) for \(n \geq 2\) is a geometric sequence with common ratio 3.

Do you have any questions or want more details?

Here are 5 related questions to expand your understanding:
1. What would \(a_5\) be in this sequence?
2. Can you express \(a_n\) in terms of \(n\) without using recursion?
3. What is the general form of a geometric sequence?
4. How would the sequence change if the multiplier was different, say 3 instead of 2?
5. Could this sequence represent any real-world scenario, such as in finance or physics?

**Tip**: To verify if a sequence is geometric, always check if the ratio \(\frac{a_{n+1}}{a_n}\) is constant across all \(n\).

The sequence {an} is defined as follows: a1 = 2, an+1 = 2(a1 + a2 + ⋅ ⋅ ⋅ + an) for n = 1, 2, 3 . . . (i) Evaluate the numerical value of a2, a3 and a4. (ii) Find the ratio of an+1/an where n ≥ 2. Is {an} a geometric sequence?

This problem explores a recursive sequence defined as a1 = 2 and an+1 = 2(a1 + a2 + ⋅ ⋅ ⋅ + an). Learn how to calculate the numerical values of a2, a3, and a4, and discover if the sequence forms a geometric sequence. Suitable for students in Grades 9-11.

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