Algebraic expressions · Ages 12–13

Patterns in sequences worksheet

Part of: Algebraic expressions & sequences

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Algebraic expressions · Ages 12–13

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Name:Date:Score:／48

Write each answer in the space. For word problems, show your working and write a sentence answer.

1.An arithmetic sequence: 4, 11, 18, 25,
Answer:
2.A geometric sequence: 3, 12, 48, 192,
Answer:
3.An arithmetic sequence: 5, 9, 13, 17,
Answer:
4.Each term is the sum of the two before: 7, 8, 15, 23, ,
Fill in each blank
Answer:
5.An arithmetic sequence, going backwards: , 6, 11, 16
Answer:
6.6-sided polygon numbers: 1, 6, 15, 28, … The gap grows by 4 each time. Number 9 is .
Answer:
7.4-sided polygon numbers: 1, 4, 9, 16, … The gap grows by 2 each time. Number 7 is .
Answer:
8.Each term is the sum of the two before: 3, 3, 6, 9, ,
Fill in each blank
Answer:
9.An arithmetic sequence, going backwards: , −12, −7, −2
Answer:
10.The sequence's rule: each term = the term before × 2 − 6. Term 1 is 4. Term 4 is .
Answer:
11.The sequence's rule: each term = the term before × 2 − 9. Term 1 is 10. Term 4 is .
Answer:

13.In a sequence, each term is the sum of the two before. Term 2 is 11 and term 5 is 43. Term 3 is and term 4 is .
Fill in each blank
Working:
Answer:
14.A bus leaves every 25 minutes. How many minutes after bus 1 does bus 12 leave?
- A. 299
- B. 300
- C. 275
- D. 325
Choose the right answer
Working:
Answer:
15.A ball falls from 136 cm, each bounce half the height before. How high is bounce 3?
Working:
Answer:
16.Climb stairs 1 or 2 steps at a time. 1 step has 1 way, and 2 steps have 2. How many ways are there for 7?
Working:
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17.Cans stack in a triangle: 13 on the bottom, one fewer on each row up. How many cans are there?
Working:
Answer:
18.Money doubles each day, then $4 goes. After day 3: $228. After day 2: $; after day 1: $; at the start: $.
Fill in each blank
Working:
Answer:

Show the answers and working

Patterns in sequences worksheet: Answers and working

1. 32
Each step is +7: 25 + 7 = 32.
2. 768
Times 4 each time: 192 × 4 = 768.
3. 21
Each step is +4: 17 + 4 = 21.
4. 38, 61
15 + 23 = 38, then 23 + 38 = 61.
5. 1
Going forwards adds 5, so going backwards takes it off: 6 − 5 = 1.
6. 153
The gaps are 5, 9, 13, then 4 more each time. Keep adding up to place 9: 153.
7. 49
The gaps are 3, 5, 7, then 2 more each time. Keep adding up to place 7: 49.
8. 15, 24
6 + 9 = 15, then 9 + 15 = 24.
9. −17
Going forwards adds 5, so going backwards takes it off: −12 − 5 = −17.
10. −10
Term by term: term 2 is 2, term 3 is −2, term 4 is −10. Each step is times, then minus.
11. 17
Term by term: term 2 is 11, term 3 is 13, term 4 is 17. Each step is times, then minus.

12. B. 5, 15, 45, 135
A geometric sequence multiplies by the same number: 5 × 3 = 15, 15 × 3 = 45, 45 × 3 = 135.

Marking: 2 for the working, 1 for the answer, 1 for the sentence.

13. 16, 27
Term 4 = term 3 + 11, and term 5 = term 3 × 2 + 11 = 43. So term 3 = (43 − 11) ÷ 2 = 16, and term 4 = 16 + 11 = 27.
14. C. 275
There are 11 gaps between the buses: 11 × 25 = 275 minutes.
15. 17
A geometric sequence, halving each time, 3 halvings in all: 136 ÷ 2³ = 17.
16. 21
The last move comes from step 6 by one, or from step 5 by two: so the ways add the two before, 13 + 8 = 21 (1, 2, 3, 5, 8, …).
17. 91
A triangle number: 1 + 2 + … + 13. 12 rows hold 78 cans, and 13 more make 91.
18. 116, 60, 32
Backwards: add 4 back, then halve: (228 + 4) ÷ 2 = 116, (116 + 4) ÷ 2 = 60, (60 + 4) ÷ 2 = 32.

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