arithmetic sequence Worksheet - WordMint - Free Printable

To solve the Arithmetic Sequences Crossword Puzzle, we need to determine the terms of each arithmetic sequence given in the clues. Let's go through each clue step by step.

Across Clues

1. 1761, 1788, 1815… 14th term?
- The sequence is: 1761, 1788, 1815, ...
- Common difference \( d = 1788 - 1761 = 27 \)
- First term \( a = 1761 \)
- The \( n \)-th term formula for an arithmetic sequence is: \( a_n = a + (n-1)d \)
- For the 14th term (\( n = 14 \)):
\[
a_{14} = 1761 + (14-1) \cdot 27 = 1761 + 13 \cdot 27 = 1761 + 351 = 2112
\]
- Answer: 2112

5. 11, 46, 81… 17th term?
- The sequence is: 11, 46, 81, ...
- Common difference \( d = 46 - 11 = 35 \)
- First term \( a = 11 \)
- For the 17th term (\( n = 17 \)):
\[
a_{17} = 11 + (17-1) \cdot 35 = 11 + 16 \cdot 35 = 11 + 560 = 571
\]
- Answer: 571

9. 8, 13, 18… 327th term?
- The sequence is: 8, 13, 18, ...
- Common difference \( d = 13 - 8 = 5 \)
- First term \( a = 8 \)
- For the 327th term (\( n = 327 \)):
\[
a_{327} = 8 + (327-1) \cdot 5 = 8 + 326 \cdot 5 = 8 + 1630 = 1638
\]
- Answer: 1638

10. 77, 85, 93… 15th term?
- The sequence is: 77, 85, 93, ...
- Common difference \( d = 85 - 77 = 8 \)
- First term \( a = 77 \)
- For the 15th term (\( n = 15 \)):
\[
a_{15} = 77 + (15-1) \cdot 8 = 77 + 14 \cdot 8 = 77 + 112 = 189
\]
- Answer: 189

12. -12, -6, 0… 43rd term?
- The sequence is: -12, -6, 0, ...
- Common difference \( d = -6 - (-12) = 6 \)
- First term \( a = -12 \)
- For the 43rd term (\( n = 43 \)):
\[
a_{43} = -12 + (43-1) \cdot 6 = -12 + 42 \cdot 6 = -12 + 252 = 240
\]
- Answer: 240

13. -3, -6, -9… 4th term?
- The sequence is: -3, -6, -9, ...
- Common difference \( d = -6 - (-3) = -3 \)
- First term \( a = -3 \)
- For the 4th term (\( n = 4 \)):
\[
a_4 = -3 + (4-1) \cdot (-3) = -3 + 3 \cdot (-3) = -3 - 9 = -12
\]
- Answer: -12

Down Clues

2. 15, 12, 9… 16th term?
- The sequence is: 15, 12, 9, ...
- Common difference \( d = 12 - 15 = -3 \)
- First term \( a = 15 \)
- For the 16th term (\( n = 16 \)):
\[
a_{16} = 15 + (16-1) \cdot (-3) = 15 + 15 \cdot (-3) = 15 - 45 = -30
\]
- Answer: -30

3. 655, 709, 763… 4th term?
- The sequence is: 655, 709, 763, ...
- Common difference \( d = 709 - 655 = 54 \)
- First term \( a = 655 \)
- For the 4th term (\( n = 4 \)):
\[
a_4 = 655 + (4-1) \cdot 54 = 655 + 3 \cdot 54 = 655 + 162 = 817
\]
- Answer: 817

4. 146, 88, 30… 29th term?
- The sequence is: 146, 88, 30, ...
- Common difference \( d = 88 - 146 = -58 \)
- First term \( a = 146 \)
- For the 29th term (\( n = 29 \)):
\[
a_{29} = 146 + (29-1) \cdot (-58) = 146 + 28 \cdot (-58) = 146 - 1624 = -1478
\]
- Answer: -1478

6. -56, -40, -24… 52nd term?
- The sequence is: -56, -40, -24, ...
- Common difference \( d = -40 - (-56) = 16 \)
- First term \( a = -56 \)
- For the 52nd term (\( n = 52 \)):
\[
a_{52} = -56 + (52-1) \cdot 16 = -56 + 51 \cdot 16 = -56 + 816 = 760
\]
- Answer: 760

7. 20, 24, 28… 50th term?
- The sequence is: 20, 24, 28, ...
- Common difference \( d = 24 - 20 = 4 \)
- First term \( a = 20 \)
- For the 50th term (\( n = 50 \)):
\[
a_{50} = 20 + (50-1) \cdot 4 = 20 + 49 \cdot 4 = 20 + 196 = 216
\]
- Answer: 216

8. 15, 9, 3… 7th term?
- The sequence is: 15, 9, 3, ...
- Common difference \( d = 9 - 15 = -6 \)
- First term \( a = 15 \)
- For the 7th term (\( n = 7 \)):
\[
a_7 = 15 + (7-1) \cdot (-6) = 15 + 6 \cdot (-6) = 15 - 36 = -21
\]
- Answer: -21

11. 225, 300, 375… 11th term?
- The sequence is: 225, 300, 375, ...
- Common difference \( d = 300 - 225 = 75 \)
- First term \( a = 225 \)
- For the 11th term (\( n = 11 \)):
\[
a_{11} = 225 + (11-1) \cdot 75 = 225 + 10 \cdot 75 = 225 + 750 = 975
\]
- Answer: 975

14. 657, 750, 843… 178th term?
- The sequence is: 657, 750, 843, ...
- Common difference \( d = 750 - 657 = 93 \)
- First term \( a = 657 \)
- For the 178th term (\( n = 178 \)):
\[
a_{178} = 657 + (178-1) \cdot 93 = 657 + 177 \cdot 93 = 657 + 16461 = 17118
\]
- Answer: 17118

15. 463, 509, 555… 12th term?
- The sequence is: 463, 509, 555, ...
- Common difference \( d = 509 - 463 = 46 \)
- First term \( a = 463 \)
- For the 12th term (\( n = 12 \)):
\[
a_{12} = 463 + (12-1) \cdot 46 = 463 + 11 \cdot 46 = 463 + 506 = 969
\]
- Answer: 969

Final Answer

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The crossword grid can be filled with the answers derived above. Each answer corresponds to the respective Across or Down clue.