Arithmetic Sequence worksheet for identifying and analyzing arithmetic progressions.

To determine whether the given sequences are arithmetic, we need to check if the difference between consecutive terms is constant. If the difference is the same for all consecutive pairs of terms, then the sequence is arithmetic. Let's analyze each sequence step by step.

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Sequence (a): 6, 9, 12, 15, ...

- Step 1: Calculate the difference between consecutive terms.
- \( 9 - 6 = 3 \)
- \( 12 - 9 = 3 \)
- \( 15 - 12 = 3 \)
- Step 2: The differences are all equal to 3.
- Conclusion: This is an arithmetic sequence with a common difference \( d = 3 \).

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Sequence (b): 7, 7.2, 7.4, 7.6, ...

- Step 1: Calculate the difference between consecutive terms.
- \( 7.2 - 7 = 0.2 \)
- \( 7.4 - 7.2 = 0.2 \)
- \( 7.6 - 7.4 = 0.2 \)
- Step 2: The differences are all equal to 0.2.
- Conclusion: This is an arithmetic sequence with a common difference \( d = 0.2 \).

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Sequence (c): 10, 16, 24, 32, ...

- Step 1: Calculate the difference between consecutive terms.
- \( 16 - 10 = 6 \)
- \( 24 - 16 = 8 \)
- \( 32 - 24 = 8 \)
- Step 2: The differences are not constant (6, 8, 8).
- Conclusion: This is not an arithmetic sequence.

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Sequence (d): 42, 21, 0, –21, ...

- Step 1: Calculate the difference between consecutive terms.
- \( 21 - 42 = -21 \)
- \( 0 - 21 = -21 \)
- \( -21 - 0 = -21 \)
- Step 2: The differences are all equal to -21.
- Conclusion: This is an arithmetic sequence with a common difference \( d = -21 \).

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Sequence (e): –2.3, –3.8, –5.3, –6.8, ...

- Step 1: Calculate the difference between consecutive terms.
- \( -3.8 - (-2.3) = -3.8 + 2.3 = -1.5 \)
- \( -5.3 - (-3.8) = -5.3 + 3.8 = -1.5 \)
- \( -6.8 - (-5.3) = -6.8 + 5.3 = -1.5 \)
- Step 2: The differences are all equal to -1.5.
- Conclusion: This is an arithmetic sequence with a common difference \( d = -1.5 \).

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Sequence (f): –1, –36, –71, –106, ...

- Step 1: Calculate the difference between consecutive terms.
- \( -36 - (-1) = -36 + 1 = -35 \)
- \( -71 - (-36) = -71 + 36 = -35 \)
- \( -106 - (-71) = -106 + 71 = -35 \)
- Step 2: The differences are all equal to -35.
- Conclusion: This is an arithmetic sequence with a common difference \( d = -35 \).

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Sequence (g): 5, 8, 13, 18, ...

- Step 1: Calculate the difference between consecutive terms.
- \( 8 - 5 = 3 \)
- \( 13 - 8 = 5 \)
- \( 18 - 13 = 5 \)
- Step 2: The differences are not constant (3, 5, 5).
- Conclusion: This is not an arithmetic sequence.

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Sequence (h): 18, 18, 18, 18, ...

- Step 1: Calculate the difference between consecutive terms.
- \( 18 - 18 = 0 \)
- \( 18 - 18 = 0 \)
- \( 18 - 18 = 0 \)
- Step 2: The differences are all equal to 0.
- Conclusion: This is an arithmetic sequence with a common difference \( d = 0 \).

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Sequence (i): –6, –12, –18, –24, ...

- Step 1: Calculate the difference between consecutive terms.
- \( -12 - (-6) = -12 + 6 = -6 \)
- \( -18 - (-12) = -18 + 12 = -6 \)
- \( -24 - (-18) = -24 + 18 = -6 \)
- Step 2: The differences are all equal to -6.
- Conclusion: This is an arithmetic sequence with a common difference \( d = -6 \).

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Sequence (j): 18, 27, 36, 45, ...

- Step 1: Calculate the difference between consecutive terms.
- \( 27 - 18 = 9 \)
- \( 36 - 27 = 9 \)
- \( 45 - 36 = 9 \)
- Step 2: The differences are all equal to 9.
- Conclusion: This is an arithmetic sequence with a common difference \( d = 9 \).

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Final Answer:

\[
\boxed{
\text{(a) Arithmetic, } d = 3; \text{ (b) Arithmetic, } d = 0.2; \text{ (c) Not arithmetic; } \text{ (d) Arithmetic, } d = -21; \text{ (e) Arithmetic, } d = -1.5; \text{ (f) Arithmetic, } d = -35; \text{ (g) Not arithmetic; } \text{ (h) Arithmetic, } d = 0; \text{ (i) Arithmetic, } d = -6; \text{ (j) Arithmetic, } d = 9
}
\]