Chapter 8 · II · Algebra

Sequences and Progressions Worksheet

11 questions Explicit and recursive rules AP and natural sums GP and fractals Tower of Hanoi

Try each question before opening its solution.

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Foundation

Explicit and recursive sequence rules · Arithmetic progressions and visualisation · 1 practice mark

Question 1

For the sequence 4, 7, 10, …, which is an explicit rule for n ≥ 1?

  1. aₙ = 4n
  2. aₙ = 3n + 1
  3. aₙ = 3n − 1
  4. aₙ = 7n
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(b) aₙ = 3n + 1

  1. The first term is 4 and the common difference is 3.
  2. aₙ = 4 + (n − 1)3 = 3n + 1.

Geometric progressions and visualisation · 1 practice mark

Question 2

The fifth term of the GP 3, 6, 12, … is:

  1. 24
  2. 48
  3. 96
  4. 30
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(b) 48

  1. a = 3 and r = 2.
  2. a₅ = ar⁴ = 3 × 16 = 48.

Tower of Hanoi · 1 practice mark

Question 3

The minimum moves for four disks in the Tower of Hanoi puzzle are:

  1. 8
  2. 12
  3. 15
  4. 16
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(c) 15

  1. To move n disks: move n − 1, move the largest disk, then move n − 1 again.
  2. Tₙ = 2Tₙ₋₁ + 1 with T₀ = 0; hence T₄ = 2⁴ − 1 = 15.

Common ratio · 1 practice mark

Question 4

The common ratio of 81, 27, 9, … is:

  1. 3
  2. −3
  3. 1/3
  4. −1/3
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(c) 1/3

  1. Divide a term by its previous term.
  2. 27/81 = 9/27 = 1/3.

Practice

Explicit and recursive sequence rules · Arithmetic progressions and visualisation · 3 practice marks

Question 5

An AP starts with 7 and has common difference −2. Give a recursive rule, an explicit rule, its tenth term, and explain how its points (n, aₙ) appear on a graph.

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a₁ = 7, aₙ₊₁ = aₙ − 2; aₙ = 9 − 2n; a₁₀ = −11.

  1. The recursive rule subtracts 2 each time.
  2. aₙ = 7 + (n − 1)(−2) = 9 − 2n; n = 10 gives −11.
  3. Only positive integer indices are allowed. Plot separate collinear points with downward slope −2, not a continuous sequence curve.

Sum of first n natural numbers · 3 practice marks

Question 6

Derive the sum of the first n natural numbers and use it to find 1 + 2 + … + 50.

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n(n + 1)/2; the sum to 50 is 1275.

  1. Write S = 1 + 2 + … + n and the same sum in reverse.
  2. Adding corresponding terms gives 2S = n(n + 1).
  3. Divide by 2; for n = 50, S = 50 × 51/2 = 1275.

Arithmetic progression · 3 practice marks

Question 7

For the AP 12, 8, 4, … write explicit and recursive rules. Find its twentieth term.

Show step-wise answer

aₙ = 16 − 4n; a₁ = 12, aₙ₊₁ = aₙ − 4; a₂₀ = −64.

  1. The common difference is −4.
  2. Use aₙ = 12 + (n − 1)(−4) = 16 − 4n for n ≥ 1.
  3. At n = 20 the term is 16 − 80 = −64. The recurrence starts from 12 and subtracts 4 each time.

Challenge

Geometric progressions and visualisation · Explicit and recursive sequence rules · Linear patterns, growth, decay and modelling · 4 practice marks

Question 8

A culture has 80 cells initially and doubles each hour. Give explicit and recursive rules, find its population after 5 hours, and contrast its plotted points with linear growth.

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Nₙ = 80 × 2ⁿ, N₀ = 80, Nₙ₊₁ = 2Nₙ; N₅ = 2560.

  1. Each step multiplies by 2, so the values form a GP indexed from n = 0.
  2. The general rule is Nₙ = 80 × 2ⁿ and the recurrence is Nₙ₊₁ = 2Nₙ.
  3. After 5 hours the model gives 80 × 32 = 2560.
  4. Differences grow rather than staying constant. Its integer-time points are not collinear, unlike a linear model.

GP in fractals · Geometric progressions and visualisation · 4 practice marks

Question 9

A line segment of length 1 is replaced at each Koch-curve step by four segments, each one third as long. After n steps, find the segment count, individual length and total length. Evaluate at n = 2.

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4ⁿ segments of length 3⁻ⁿ; total length (4/3)ⁿ. At n = 2: 16 segments, each 1/9, total 16/9.

  1. Each stage multiplies the segment count by 4, starting with one segment.
  2. Each segment length is multiplied by 1/3, starting at length 1.
  3. Multiply count and length: 4ⁿ × 3⁻ⁿ = (4/3)ⁿ.
  4. Substitute n = 2. This illustrates GP scaling in a fractal construction.

Tower of Hanoi · Explicit and recursive sequence rules · 4 practice marks

Question 10

Explain the recursive Tower of Hanoi algorithm for n disks and derive its minimum-move formula. Calculate it for 5 disks.

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T₀ = 0; Tₙ = 2Tₙ₋₁ + 1 = 2ⁿ − 1; 31 moves for 5 disks.

  1. Move the top n − 1 disks to the spare peg using the same rules.
  2. Move the largest disk to the destination; then move the n − 1 disks onto it.
  3. This is necessary as well as sufficient: the largest disk can move only after the smaller disks are cleared, and they must be moved back. Thus Tₙ = 2Tₙ₋₁ + 1.
  4. Tₙ + 1 = 2(Tₙ₋₁ + 1); starting at T₀ + 1 = 1 gives Tₙ = 2ⁿ − 1. T₅ = 31.

Recursive and explicit Hanoi rules · 4 practice marks

Question 11

A learner proposes Tₙ = 2n − 1 as the minimum move count for n Hanoi disks. Use n = 3 to disprove it and obtain the correct recursive and explicit rules.

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The correct rules are T₀ = 0, Tₙ = 2Tₙ₋₁ + 1 and Tₙ = 2ⁿ − 1.

  1. For three disks the minimum is 7 moves, whereas 2n − 1 gives 5.
  2. Move the n − 1 smaller disks away, move the largest once, then transfer the smaller disks onto it.
  3. Thus Tₙ = 2Tₙ₋₁ + 1 with T₀ = 0.
  4. Adding 1 gives Tₙ + 1 = 2(Tₙ₋₁ + 1), so Tₙ + 1 = 2ⁿ and Tₙ = 2ⁿ − 1.