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Let's solve this step by step.

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### 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:
\[
4, 20, 100
\]

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### 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:
\[
T_n = 4 \times 5^{(n-1)}
\]

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

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

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### 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!

The term-to-term rule for a sequence is to start at 4 and multiply by 5 each time.
a) Work out the first three terms of the sequence.
b) Write the nth term rule for the sequence in the form a × r^(n-1).

Learn how to solve a geometric sequence where the term-to-term rule starts at 4 and multiplies by 5 each time. This step-by-step guide calculates the first three terms and derives the nth term formula in the form a × r^(n-1), suitable for students in Grades 8-10.

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