Printable · GCSE Higher · ages 14-16
Special sequences: square, cube, Fibonacci, quadratic, geometric worksheet — GCSE Higher
Fifteen questions on "special sequences: square, cube, fibonacci, quadratic, geometric" — DfE statement A24. Print it, or print three versions so neighbours cannot copy by letter; the key gives the letter for each version.
part Higher
Special sequences: square, cube, Fibonacci, quadratic, geometric worksheet — GCSE Higher
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- 1.A tree is 2 metres tall. Each year its height increases by 10% of its height at the start of that year. Work out the height of the tree after 3 years, giving your answer to 1 decimal place.
- 2.A colony of bacteria doubles in number every hour. At 9am there are 5 bacteria in the colony. Work out how many bacteria there will be at 12 noon.
- 3.The nth term of a geometric sequence is given by aₙ = 3 × 2ⁿ⁻¹. Work out the position of the first term that is greater than 1,000.
- 4.A Fibonacci-type sequence begins 3, 5, 8, 13, ... where each term after the second is the sum of the two terms before it. Work out the 7th term of the sequence.
- 5.A geometric sequence has first term 3 and common ratio √2. Work out the 6th term of the sequence, giving your answer in the form a√2.
- 6.An arithmetic sequence has first term 40 and common difference −6. Work out the 9th term of the sequence.
- 7.A fundraising chain letter starts with 3 people taking part in round 1. Each following round has 4 times as many people taking part as the round before. Work out the number of people taking part in round 4.
- 8.A Fibonacci-type sequence is defined by aₙ = aₙ₋₁ + aₙ₋₂ for n ≥ 3. Given that a₁ = 5 and a₃ = 17, work out a₂.
- 9.Find the common ratio of the geometric sequence 5, 5√2, 10, 10√2, ... Give your answer in surd form.
- 10.A geometric sequence begins 162, 54, 18, 6, ... Work out the next term in the sequence.
- 11.The 2nd term of a geometric sequence is 6 and the 4th term is 24. Given that all the terms of the sequence are positive, work out the common ratio.
- 12.Four sequences are shown below. Sequence P: 3, 6, 12, 24, … Sequence Q: 1, 4, 9, 16, … Sequence R: 2, 4, 8, 14, … Sequence S: 5, 10, 15, 20, … Work out which of the sequences P, Q, R and S is geometric.
- 13.A geometric sequence begins 80, 40, 20, 10, ... Work out the next term in the sequence.
- 14.A geometric sequence has first term 3 and common ratio 2. Work out the 5th term of the sequence.
- 15.A geometric sequence has first term 4 and common ratio √2. Work out the sum of the first three terms of the sequence, giving your answer in the form a + b√2.
Answer key
- (d) 2.7 — The height after 3 years of 10% compound growth is 2 × 1.1³ = 2.662 m, which rounds to 2.7 m. A candidate who adds 10% of the original height (0.2 m) in each of the 3 years, instead of compounding on the new height each time, would reach 2 + 3×0.2 = 2.6 m. A candidate who compounds for only 2 years would reach 2 × 1.1² = 2.42 m, rounding to 2.4 m. A candidate who compounds for 4 years instead of 3 would reach 2 × 1.1⁴ = 2.928 m, rounding to 2.9 m.
- (a) 40 — From 9am to 12 noon is 3 hours, so the population doubles three times: 5 × 2³ = 40. A candidate who counts the elapsed time as 4 hours (an off-by-one counting error) would reach 5 × 2⁴ = 80. A candidate who counts it as only 2 hours would reach 5 × 2² = 20. A candidate who misreads 'doubles' as 'increases by 2' each hour would compute 5 + 3 × 2 = 11.
- (b) 10 — Check n = 9: a₉ = 3 × 2⁸ = 3 × 256 = 768, below 1,000. Check n = 10: a₁₀ = 3 × 2⁹ = 3 × 512 = 1,536, above 1,000. So the first term greater than 1,000 is at position n = 10. Answering 9 comes from forgetting the −1 shift and using the formula as 3 × 2ⁿ instead of 3 × 2ⁿ⁻¹: checking 3 × 2⁹ = 1,536 (which exceeds 1,000) but then reporting the position as n = 9, the exponent used, instead of n = 10 — wrong, because the exponent in the real formula is n − 1, not n. Answering 11 comes from going one term too far: correctly finding that a₁₀ already exceeds 1,000, but then checking one position further and reporting n = 11 instead of stopping at the first position that already works — wrong, because n = 11 is not the FIRST term greater than 1,000. Answering 1,536 gives the VALUE of the term (a₁₀ itself) rather than its position — wrong, because the question asks which term it is (the value of n), not what that term is worth.
- (c) 55 — Continuing the pattern: 8+13=21 (5th term), 13+21=34 (6th term), 21+34=55 (7th term). A candidate who miscounts the position and stops one term early would give 34, the 6th term. A candidate who doubles the most recent term instead of adding the two before it would compute 34×2=68. A candidate who adds a non-adjacent pair — the 4th and 6th terms, skipping the 5th — would compute 13+34=47.
- (d) 12√2 — The nth term of a geometric sequence is a × rⁿ⁻¹. Here a = 3, r = √2, n = 6, so the 6th term is 3 × (√2)⁵. Since (√2)² = 2, (√2)⁴ = (2)² = 4, so (√2)⁵ = (√2)⁴ × √2 = 4√2. The 6th term is 3 × 4√2 = 12√2. 15√2 comes from treating (√2)⁵ as 5√2 — multiplying the index by the surd instead of raising √2 to that power — then multiplying by 3 gives 3 × 5√2 = 15√2, which is wrong because powers of a surd do not scale linearly with the index. 24√2 comes from miscalculating (√2)⁴ as 8 instead of 4 (a squaring slip, since (√2)² = 2 but (√2)⁴ should be 2² = 4, not 2 × 4), giving 3 × 8√2 = 24√2. 4√2 comes from forgetting to multiply by the first term a = 3, leaving just (√2)⁵ = 4√2.
- (c) −8 — The nth term is the first term plus (n − 1) lots of the common difference: 40 + 8 × (−6) = 40 − 48 = −8. A candidate who uses 9 lots of the common difference instead of 8 gets 40 + 9 × (−6) = −14. A candidate who treats the common difference as +6 instead of −6 gets 40 + 8 × 6 = 88. A candidate who uses only 7 lots of the common difference gets 40 + 7 × (−6) = −2.
- (b) 192 — This is a geometric sequence with first term 3 and common ratio 4: round 2 has 3 × 4 = 12, round 3 has 12 × 4 = 48, round 4 has 48 × 4 = 192. A candidate who applies the ×4 multiplier four times instead of three gets 3 × 4⁴ = 768. A candidate who wrongly treats the growth as arithmetic, taking the round 2 figure of 12 as a fixed amount added each round, gets 3, 15, 27, 39. A candidate who forgets the starting 3 people and just works out 4⁴ gets 256.
- (d) 12 — Since a₃ = a₂ + a₁, rearranging gives a₂ = a₃ − a₁ = 17 − 5 = 12. 22 comes from adding instead of subtracting: a₃ + a₁ = 17 + 5 = 22. 11 comes from treating the sequence as if it were arithmetic and averaging the two given terms: (5 + 17) ÷ 2 = 11 — but a Fibonacci-type sequence isn't arithmetic, so this doesn't apply. −12 comes from subtracting in the wrong order: a₁ − a₃ = 5 − 17 = −12.
- (b) √2 — 5√2 ÷ 5 = √2. Checking between the third and second terms: 10 ÷ 5√2 = √2 as well (since 10 ÷ 5√2 = 2 ÷ √2 = √2), so the common ratio is confirmed as √2 throughout. Squaring the ratio instead of leaving it as a surd gives 2, which is wrong because 2 is the SQUARE of the common ratio, not the ratio itself. Rounding the exact surd to a decimal gives 1.41, which is wrong because the sequence is defined using an exact surd ratio, and a rounded decimal is not the same value. Writing down the SECOND TERM of the sequence, 5√2, instead of the ratio BETWEEN terms, gives 5√2, which is wrong because a term of the sequence is not the same thing as the common ratio.
- (a) 2 — The common ratio is 54 ÷ 162 = 1/3, confirmed by 18 ÷ 54 = 1/3 and 6 ÷ 18 = 1/3. The next term is 6 × 1/3 = 2. A candidate who wrongly treats the sequence as arithmetic and subtracts the last difference (18 − 6 = 12) again gets 6 − 12 = −6. A candidate who multiplies by 3 instead of dividing, effectively reversing the sequence, gets 18. A candidate who guesses the ratio is 1/2 gets 6 × 1/2 = 3.
- (c) 2 — For a geometric sequence, a₄ = a₂ × r². So r² = a₄ ÷ a₂ = 24 ÷ 6 = 4, giving r = 2 or r = −2. Since all the terms are positive, r = 2. 4 comes from computing a₄ ÷ a₂ = 4 and stopping there, without taking the square root to find r. 18 comes from subtracting instead of dividing: a₄ − a₂ = 24 − 6 = 18, taken directly as the common ratio. −2 is the other square root of 4, but it is rejected because all the sequence's terms are given as positive.
- (a) P — A sequence is geometric when consecutive terms share a constant ratio. For P: 6 ÷ 3 = 2, 12 ÷ 6 = 2, 24 ÷ 12 = 2 — the ratio is constant at 2, so P is geometric. Q is the square numbers (1², 2², 3², 4²), a quadratic sequence: its ratios are 4, 2.25, 1.78, … — not constant. R looks geometric at first (2, 4, 8 doubles each time), but the pattern breaks: 8 to 14 is a ratio of 1.75, not 2. Its first differences are 2, 4, 6 — a constant second difference of 2 — so R is a quadratic sequence, not geometric. S has a constant DIFFERENCE of 5 (it is arithmetic), but its ratios (2, 1.5, 1.33, …) are not constant, so it is not geometric.
- (d) 5 — Each term is found by multiplying the previous term by the common ratio, 0.5: 80, 40, 20, 10, and the next term is 10 × 0.5 = 5. A candidate who instead subtracts the same amount each time (repeating the last difference of 10) would reach 10 − 10 = 0. A candidate who divides by 4 instead of by 2 would reach 10 ÷ 4 = 2.5. A candidate who multiplies by 2 instead of dividing (reversing the direction of the sequence) would reach 10 × 2 = 20.
- (d) 48 — Each term is found by multiplying the previous term by the common ratio, 2: 3, 6, 12, 24, 48 — the 5th term is 48. A candidate who mistakes the common ratio for a common difference, and adds 2 four times, would reach 3+4×2=11. A candidate who works out the multiplier 2⁴=16 but forgets to multiply it by the first term would give 16. A candidate who multiplies one time too many (finding the 6th term instead of the 5th) would reach 3×2⁵=96.
- (a) 12 + 4√2 — a₁ = 4, a₂ = 4√2, and a₃ = a₁ × r² = 4 × (√2)² = 4 × 2 = 8. The sum of the first three terms is 4 + 4√2 + 8 = 12 + 4√2. 8 + 4√2 comes from leaving out a₁ and adding only a₂ + a₃ = 4√2 + 8. 36 + 4√2 comes from squaring the whole second term instead of applying the ratio to the first term: (4√2)² = 32 used as a₃, giving 4 + 4√2 + 32 = 36 + 4√2. 4 + 12√2 comes from using r³ instead of r² for the third term: 4 × (√2)³ = 4 × 2√2 = 8√2, giving 4 + 4√2 + 8√2 = 4 + 12√2.
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