This set has twelve original questions, ordered from easier to harder, covering all five lessons in sequences and pattern rules. Questions 1 to 4 are warm-ups, 5 to 8 build testing and type-spotting, and 9 to 12 mix skills.
Attempt each question on paper, without a calculator, and write your working as you would in an exam. Only then open the answer.
Mark the ones you got wrong and use the routing list at the end. The sequence and series laboratory can generate terms so you can check your rules afterwards.
Questions
1. Write the next two terms of 6, 10, 14, 18, … and find the nth term.
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The difference is 4, so the next terms are 22 and 26. The rule starts with 4n, which gives 4, 8, 12, 16. Each term is 2 more, so the rule is 4n + 2.
Check: n = 1 gives 6 and n = 4 gives 18.
2. Find the nth term of 25, 22, 19, 16, …
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The difference is −3, so the rule starts with −3n, which gives −3, −6, −9, −12. Each term is 28 more, so the rule is −3n + 28.
Check: n = 1 gives 25 and n = 4 gives 16.
3. Find the 40th term of 7, 13, 19, 25, …
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The difference is 6, so the rule starts with 6n, which gives 6, 12, 18, 24. Each term is 1 more, so the rule is 6n + 1. The 40th term is 6 × 40 + 1 = 241.
Check by counting on: 7 + 39 × 6 = 7 + 234 = 241.
4. The nth term of a sequence is 3n + 2, giving 5, 8, 11, 14, … Is 200 a term? Is 300 a term?
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For 200: 3n + 2 = 200, so 3n = 198 and n = 66. This is a whole number, so 200 is the 66th term.
For 300: 3n + 2 = 300, so 3n = 298 and n = 99.33… This is not a whole number, so 300 is not a term.
5. A student says the nth term of 4, 7, 10, 13, … is 3n + 4. Use one term to show the rule is wrong, then give the correct rule.
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At n = 1 the rule gives 3 × 1 + 4 = 7, but the first term is 4. So the rule fails. The student used the first term as the constant.
The difference is 3, and 3n gives 3, 6, 9, 12. Each term is 1 more, so the correct rule is 3n + 1.
Check: n = 4 gives 13. The 30th term is 3 × 30 + 1 = 91.
6. The sequence 162, 54, 18, 6, … continues. State its type, then find the 6th term.
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Differences: −108, −36, −12, not constant. Ratios: 54 ÷ 162 = 1/3, 18 ÷ 54 = 1/3, 6 ÷ 18 = 1/3. The ratio is constant, so it is geometric with a = 162 and r = 1/3.
The 6th term is 162 × (1/3)^5 = 162 ÷ 243 = 2/3.
Check by continuing: 6 × 1/3 = 2, then 2 × 1/3 = 2/3.
7. A geometric sequence begins 5, 15, 45, … Which term equals 1215?
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a = 5 and r = 3, so the nth term is 5 × 3^(n − 1). Set 5 × 3^(n − 1) = 1215. Then 3^(n − 1) = 243.
Since 3^5 = 243, n − 1 = 5 and n = 6. So 1215 is the 6th term.
Check by continuing: 45, 135, 405, 1215. That is the 6th term.
8. Find the nth term of 2, 7, 14, 23, 34, …
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First differences: 5, 7, 9, 11. Second differences: 2, 2, 2. So a = 1 and the rule starts with n².
Subtract n² (1, 4, 9, 16, 25): the remainders are 1, 3, 5, 7, 9, which is 2n − 1. The rule is n² + 2n − 1.
Check: n = 1 gives 2 and n = 5 gives 25 + 10 − 1 = 34.
9. Find the nth term of 6, 15, 28, 45, … and then the 10th term.
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First differences: 9, 13, 17. Second differences: 4, 4. Half of 4 is 2, so the rule starts with 2n².
Subtract 2n² (2, 8, 18, 32): the remainders are 4, 7, 10, 13, which is 3n + 1. The rule is 2n² + 3n + 1.
Check: n = 4 gives 32 + 12 + 1 = 45. The 10th term is 200 + 30 + 1 = 231.
10. The nth term of a sequence is n² − 4. Which term equals 77? Is 50 a term?
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For 77: n² − 4 = 77, so n² = 81 and n = 9 (n must be positive). So 77 is the 9th term.
For 50: n² − 4 = 50, so n² = 54. Since 7² = 49 and 8² = 64, n is not a whole number, so 50 is not a term.
11. A row of squares is made from sticks. One square uses 4 sticks, two squares joined in a row use 7, and three use 10. Find the rule, the number of sticks for 25 squares, and how many squares use 100 sticks.
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The difference is 3, so the rule starts with 3n, which gives 3, 6, 9. Each term is 1 more, so the rule is 3n + 1.
For 25 squares: 3 × 25 + 1 = 76 sticks.
For 100 sticks: 3n + 1 = 100, so 3n = 99 and n = 33 squares.
12. The first three terms of a sequence are 3, 6, 12. Rule A is “double each time”. Rule B is 1.5n² − 1.5n + 3. Show that both fit the three terms, then find the 4th term from each.
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Rule B: n = 1 gives 1.5 − 1.5 + 3 = 3. n = 2 gives 6 − 3 + 3 = 6. n = 3 gives 13.5 − 4.5 + 3 = 12. Rule A gives 3, 6, 12 by doubling. Both fit.
4th term: Rule A gives 12 × 2 = 24. Rule B gives 1.5 × 16 − 1.5 × 4 + 3 = 24 − 6 + 3 = 21.
Three terms cannot decide between the rules, so a stated assumption or extra information is needed.
If you got these wrong
- Questions 1 to 3 and 11 (rule or constant wrong): revisit finding a linear term rule from differences. Check that your constant passes the n = 1 test.
- Question 5 and 12 (rule fits some terms only): revisit testing a term rule against distant terms.
- Questions 6 and 7 (add or multiply confusion, wrong power): revisit distinguishing arithmetic from geometric change.
- Questions 8 and 9 (second difference or remainder): revisit using second differences for a quadratic rule. The quadratic structure explorer can show the link between the second difference and n².
- Questions 4, 10 and 11 (value put in place of n): revisit term position and term value.
Record each wrong question and retry a fresh one in a few days, for example with the mistake log and retest queue. The non-calculator working trainer helps with the arithmetic steps.
If one error type keeps coming back, online one-to-one Mathematics tuition lets a teacher look at your written working and trace it to the habit behind it.