Examples of arithmetic sequences with duck illustrations.

To solve the problem of finding the \( n \)-th term of each sequence, we need to analyze the pattern in each sequence and derive a general formula. Let's go through each sequence step by step.

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Sequence (a): \( 5, 9, 13, 17, \ldots \)

1. Identify the pattern:
- The difference between consecutive terms is:
\[
9 - 5 = 4, \quad 13 - 9 = 4, \quad 17 - 13 = 4
\]
- The sequence is an arithmetic sequence with a common difference \( d = 4 \).

2. General formula for an arithmetic sequence:
\[
a_n = a_1 + (n-1)d
\]
where \( a_1 \) is the first term and \( d \) is the common difference.

3. Substitute the values:
- \( a_1 = 5 \)
- \( d = 4 \)
\[
a_n = 5 + (n-1) \cdot 4
\]

4. Simplify:
\[
a_n = 5 + 4n - 4 = 4n + 1
\]

Answer for (a):
\[
\boxed{4n + 1}
\]

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Sequence (b): \( 8, 12, 16, 20, \ldots \)

1. Identify the pattern:
- The difference between consecutive terms is:
\[
12 - 8 = 4, \quad 16 - 12 = 4, \quad 20 - 16 = 4
\]
- The sequence is an arithmetic sequence with a common difference \( d = 4 \).

2. General formula for an arithmetic sequence:
\[
a_n = a_1 + (n-1)d
\]
where \( a_1 \) is the first term and \( d \) is the common difference.

3. Substitute the values:
- \( a_1 = 8 \)
- \( d = 4 \)
\[
a_n = 8 + (n-1) \cdot 4
\]

4. Simplify:
\[
a_n = 8 + 4n - 4 = 4n + 4
\]

Answer for (b):
\[
\boxed{4n + 4}
\]

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Sequence (c): \( 10, 13, 16, 19, \ldots \)

1. Identify the pattern:
- The difference between consecutive terms is:
\[
13 - 10 = 3, \quad 16 - 13 = 3, \quad 19 - 16 = 3
\]
- The sequence is an arithmetic sequence with a common difference \( d = 3 \).

2. General formula for an arithmetic sequence:
\[
a_n = a_1 + (n-1)d
\]
where \( a_1 \) is the first term and \( d \) is the common difference.

3. Substitute the values:
- \( a_1 = 10 \)
- \( d = 3 \)
\[
a_n = 10 + (n-1) \cdot 3
\]

4. Simplify:
\[
a_n = 10 + 3n - 3 = 3n + 7
\]

Answer for (c):
\[
\boxed{3n + 7}
\]

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Sequence (d): \( 21, 19, 17, 15, \ldots \)

1. Identify the pattern:
- The difference between consecutive terms is:
\[
19 - 21 = -2, \quad 17 - 19 = -2, \quad 15 - 17 = -2
\]
- The sequence is an arithmetic sequence with a common difference \( d = -2 \).

2. General formula for an arithmetic sequence:
\[
a_n = a_1 + (n-1)d
\]
where \( a_1 \) is the first term and \( d \) is the common difference.

3. Substitute the values:
- \( a_1 = 21 \)
- \( d = -2 \)
\[
a_n = 21 + (n-1) \cdot (-2)
\]

4. Simplify:
\[
a_n = 21 - 2(n-1) = 21 - 2n + 2 = 23 - 2n
\]

Answer for (d):
\[
\boxed{23 - 2n}
\]

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Sequence (e): \( 4, -1, -6, -11, \ldots \)

1. Identify the pattern:
- The difference between consecutive terms is:
\[
-1 - 4 = -5, \quad -6 - (-1) = -5, \quad -11 - (-6) = -5
\]
- The sequence is an arithmetic sequence with a common difference \( d = -5 \).

2. General formula for an arithmetic sequence:
\[
a_n = a_1 + (n-1)d
\]
where \( a_1 \) is the first term and \( d \) is the common difference.

3. Substitute the values:
- \( a_1 = 4 \)
- \( d = -5 \)
\[
a_n = 4 + (n-1) \cdot (-5)
\]

4. Simplify:
\[
a_n = 4 - 5(n-1) = 4 - 5n + 5 = 9 - 5n
\]

Answer for (e):
\[
\boxed{9 - 5n}
\]

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Sequence (f): \( \frac{1}{2}, 1, \frac{3}{2}, 2, \frac{5}{2}, 3, \ldots \)

1. Identify the pattern:
- Convert all terms to fractions for consistency:
\[
\frac{1}{2}, \frac{2}{2}, \frac{3}{2}, \frac{4}{2}, \frac{5}{2}, \frac{6}{2}, \ldots
\]
- The numerator increases by 1 each time, and the denominator is always 2.
- The \( n \)-th term can be written as:
\[
a_n = \frac{n}{2}
\]

2. Verify:
- For \( n = 1 \): \( a_1 = \frac{1}{2} \)
- For \( n = 2 \): \( a_2 = \frac{2}{2} = 1 \)
- For \( n = 3 \): \( a_3 = \frac{3}{2} \)
- This matches the given sequence.

Answer for (f):
\[
\boxed{\frac{n}{2}}
\]

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Final Answers:

\[
\boxed{4n + 1, 4n + 4, 3n + 7, 23 - 2n, 9 - 5n, \frac{n}{2}}
\]