To find the general term (nth term) of a sequence, we follow these steps:

1. Identify the pattern – Determine if the sequence is arithmetic, geometric, or another type.

2. Find the first term ($a_1$).

3. Determine the common difference ($d$) – This applies to arithmetic sequences.

4. Use the formula for the nth term of an arithmetic sequence:

   $a_n = a_1 + (n-1) d$

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Solution for (i): $4, 9, 14, 19, ...$

1. First term ($a_1$) = 4

2. Common difference ($d$) = $9 - 4 = 5$

3. Use the formula:

   $a_n = 4 + (n - 1) \cdot 5$

4. Simplify:

   $a_n = 4 + 5n - 5$

   $a_n = 5n - 1$

General term: $a_n = 5n - 1$

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Solution for (ii): $25, 20, 15, 10, ...$

1. First term ($a_1$) = 25

2. Common difference ($d$) = $20 - 25 = -5$

3. Use the formula:

   $a_n = 25 + (n - 1) (-5)$

4. Simplify:

   $a_n = 25 - 5n + 5$

   $a_n = -5n + 30$

General term: $a_n = -5n + 30$

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Final Answers:

(i) $a_n = 5n - 1$
(ii) $a_n = -5n + 30$

Would you like a step-by-step verification with examples?

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Related Questions:

1. How do you derive the formula for an arithmetic sequence?
2. What is the general term for a geometric sequence?
3. Can you explain the difference between arithmetic and geometric sequences?
4. How do you find the sum of the first $n$ terms of an arithmetic sequence?
5. What happens if the common difference is zero?

Tip:

To check your general term, substitute $n = 1, 2, 3,$ etc., and see if you get the original sequence!