Let's solve this step by step.

Part (a): Work out the first three terms of the sequence.

The term-to-term rule states:

• Start at 4 and multiply by 5 each time.

The first term is: $T_1 = 4$

The second term is: $T_2 = 4 \times 5 = 20$

The third term is: $T_3 = 20 \times 5 = 100$

Thus, the first three terms are: $4, 20, 100$

Part (b): Write the $n^{\text{th}}$ term rule in the form $a \times r^{(n-1)}$.

The general form for a geometric sequence is: $T_n = a \times r^{(n-1)}$ where:

• $a$ is the first term,
• $r$ is the common ratio,
• $n$ is the term number.

From the problem:

• $a = 4$ (first term),
• $r = 5$ (common ratio).

Substitute these into the formula: $T_n = 4 \times 5^{(n-1)}$

Final Answers:

a) The first three terms are: 4, 20, 100.
b) The $n^{\text{th}}$ term rule is: $T_n = 4 \times 5^{(n-1)}.$

Do you need further clarification on any part? Let me know!

Related Questions:

1. What are the first five terms of the sequence?
2. How can you verify that a sequence is geometric?
3. What is the 6th term of this sequence?
4. How do you calculate the sum of the first $n$ terms of a geometric sequence?
5. What happens to the sequence if the common ratio is less than 1?

Tip: In geometric sequences, the ratio between consecutive terms remains constant, which makes it easy to identify the next terms!