Mathematics · Simulation and virtual lab Free
Sequences and series: savings, bouncing balls and growth
Add terms one at a time to savings, a bouncing ball and growing bacteria, and watch the sums grow or settle to a limit.
- You will learn to
- Recognise arithmetic and geometric sequences and find the nth term.
- Find the sum of n terms of each type.
- Decide when a geometric series has a sum to infinity and find it.
- You will measure
- uₙ and Sₙ for each n; the sum to infinity when it exists.
- Time needed
- About 25–35 min
- Equipment
- Nothing extra. Real version: a ball and metre rule (measure bounce heights), squared paper.
- Price
- Free: no payment, no account.
Virtual lab
Follow the steps with the simulation above (go back to the simulation ↑). Each time you tap Record, a row is added to the table and a point to the graph.
Aim
To recognise arithmetic and geometric sequences, to find and use the nth term and the sum of n terms, and to investigate when a geometric series has a sum to infinity.
What to do
- Choose SACCO savings: first deposit 2 000 RWF, extra 500 RWF each month. Tap Next term several times. Write the nth term.
- Calculate the total after 12 months with Sₙ = n ÷ 2 × (2a + (n − 1)d), then check with Add 5 terms.
- Look at the graph: which points lie on a straight line, the terms or the sums?
- Choose the bouncing ball with r = 0.6. Add terms and watch the purple bar. What number does the sum approach?
- Change r to 0.8 and 0.4. Check the limit with S∞ = a ÷ (1 − r).
- Choose bacteria with r = 2. Does the sum have a limit? Explain why.
Results
Questions
Is 3, 7, 11, 15, … arithmetic or geometric? Find its 20th term.
Arithmetic with a = 3, d = 4: u₂₀ = 3 + 19 × 4 = 79.
Find the sum of the first 12 months of savings: 2 000 RWF then 500 RWF more each month.
S₁₂ = 12 ÷ 2 × (2 × 2 000 + 11 × 500) = 6 × 9 500 = 57 000 RWF.
A ball bounces to 2 m, then each bounce is 0.6 of the one before. How high is the 5th bounce?
u₅ = 2 × 0.6⁴ = 0.259 m.
What is the sum to infinity of the bounce heights 2 + 1.2 + 0.72 + …?
S∞ = 2 ÷ (1 − 0.6) = 5 m.
Why does a geometric series with r = 2 have no sum to infinity?
Each term is bigger than the one before, so the sum keeps growing without limit. A sum to infinity exists only when −1 < r < 1.
Five bacteria double every hour. How many are there after 10 hours (at the start of the 11th hour)?
u₁₁ = 5 × 2¹⁰ = 5 120.
For teachers: expected results and how the model works
Arithmetic: uₙ = a + (n − 1)d, Sₙ = n ÷ 2 × (2a + (n − 1)d). Geometric: uₙ = a rⁿ⁻¹, Sₙ = a(1 − rⁿ) ÷ (1 − r), S∞ = a ÷ (1 − r) for |r| < 1. Default values: savings a = 2 000, d = 500: u₁₂ = 7 500, S₁₂ = 57 000 RWF (up to 24 months). Ball: first bounce 2 m, r = 0.6: S∞ = 5 m (sliders 0.5–3 m and r 0.3–0.9, up to 16 bounces). Bacteria: a = 5, r = 2: u₁₀ = 2 560, S₁₀ = 5 115 (up to 12 hours). Each new term is added to the table automatically; Undo removes the last term. In the ball picture each arc is one bounce and the dashed line shows its height. Compound interest (S3 Unit 7) is a geometric sequence with r = 1 + rate.
Exam practice Exam set
National-exam style questions on this simulation (20 marks). Try each one on paper first, then open the marking scheme.
The 3rd term of an arithmetic sequence is 11 and the 8th term is 31. Find (a) the common difference, (b) the first term, (c) the sum of the first 20 terms. (6 marks)
Marking scheme: (a) a + 2d = 11, a + 7d = 31 (1); 5d = 20, d = 4 (1). (b) a = 3 (1). (c) S₂₀ = 10 × (6 + 19 × 4) (2) = 820 (1).
A geometric sequence has first term 48 and common ratio ½. (a) Find the 6th term. (b) Find the sum of the first 6 terms. (c) Find the sum to infinity. (6 marks)
Marking scheme: (a) u₆ = 48 × (½)⁵ = 1.5 (2). (b) S₆ = 48(1 − (½)⁶) ÷ (1 − ½) (1) = 94.5 (1). (c) S∞ = 48 ÷ ½ = 96 (2).
A ball is dropped from 3 m and each bounce reaches ¾ of the previous height. Find the total distance travelled before it stops. (4 marks)
Marking scheme: Bounce heights 2.25, 1.6875, … with r = ¾ (1); sum of bounce heights = 2.25 ÷ ¼ = 9 m (1); each bounce is travelled up and down: 2 × 9 = 18 m (1); total = 3 + 18 = 21 m (1).
A SACCO pays 5 % interest per year. 100 000 RWF is invested. After how many complete years will it first exceed 150 000 RWF? (4 marks)
Marking scheme: 100 000 × 1.05ⁿ > 150 000 (1); 1.05ⁿ > 1.5 (1); n > log 1.5 ÷ log 1.05 = 8.3 (1); 9 years (1).
Donat Sciences and MathsMathematics · Virtual lab worksheet
School: ........................................ Name: ................................ Class: ........ Date: ............
Sequences and series: savings, bouncing balls and growth
Aim: To recognise arithmetic and geometric sequences, to find and use the nth term and the sum of n terms, and to investigate when a geometric series has a sum to infinity.
Apparatus (real laboratory): a ball and metre rule (measure bounce heights), squared paper.
What to do (with the simulation on www.donatstem.co.rw)
- Choose SACCO savings: first deposit 2 000 RWF, extra 500 RWF each month. Tap Next term several times. Write the nth term.
- Calculate the total after 12 months with Sₙ = n ÷ 2 × (2a + (n − 1)d), then check with Add 5 terms.
- Look at the graph: which points lie on a straight line, the terms or the sums?
- Choose the bouncing ball with r = 0.6. Add terms and watch the purple bar. What number does the sum approach?
- Change r to 0.8 and 0.4. Check the limit with S∞ = a ÷ (1 − r).
- Choose bacteria with r = 2. Does the sum have a limit? Explain why.
Results
| # | n | uₙ | Sₙ | Formula check |
|---|
| 1 | | | | |
| 2 | | | | |
| 3 | | | | |
| 4 | | | | |
| 5 | | | | |
| 6 | | | | |
| 7 | | | | |
| 8 | | | | |
| 9 | | | | |
| 10 | | | | |
| 11 | | | | |
| 12 | | | | |
Graph
Plot uₙ and Sₙ against n on the same axes for one arithmetic and one geometric sequence. Describe each shape.
Questions
- Is 3, 7, 11, 15, … arithmetic or geometric? Find its 20th term. [2]
- Find the sum of the first 12 months of savings: 2 000 RWF then 500 RWF more each month. [2]
- A ball bounces to 2 m, then each bounce is 0.6 of the one before. How high is the 5th bounce? [2]
- What is the sum to infinity of the bounce heights 2 + 1.2 + 0.72 + …? [2]
- Why does a geometric series with r = 2 have no sum to infinity? [2]
- Five bacteria double every hour. How many are there after 10 hours (at the start of the 11th hour)? [2]
- The 3rd term of an arithmetic sequence is 11 and the 8th term is 31. Find (a) the common difference, (b) the first term, (c) the sum of the first 20 terms. [6]
- A geometric sequence has first term 48 and common ratio ½. (a) Find the 6th term. (b) Find the sum of the first 6 terms. (c) Find the sum to infinity. [6]
- A ball is dropped from 3 m and each bounce reaches ¾ of the previous height. Find the total distance travelled before it stops. [4]
- A SACCO pays 5 % interest per year. 100 000 RWF is invested. After how many complete years will it first exceed 150 000 RWF? [4]
Donat Sciences and MathsMathematics · Virtual lab worksheet
School: ........................................ Name: ................................ Class: ........ Date: ............
Marking scheme: Sequences and series: savings, bouncing balls and growth
Expected results
Arithmetic: uₙ = a + (n − 1)d, Sₙ = n ÷ 2 × (2a + (n − 1)d). Geometric: uₙ = a rⁿ⁻¹, Sₙ = a(1 − rⁿ) ÷ (1 − r), S∞ = a ÷ (1 − r) for |r| < 1. Default values: savings a = 2 000, d = 500: u₁₂ = 7 500, S₁₂ = 57 000 RWF (up to 24 months). Ball: first bounce 2 m, r = 0.6: S∞ = 5 m (sliders 0.5–3 m and r 0.3–0.9, up to 16 bounces). Bacteria: a = 5, r = 2: u₁₀ = 2 560, S₁₀ = 5 115 (up to 12 hours). Each new term is added to the table automatically; Undo removes the last term. In the ball picture each arc is one bounce and the dashed line shows its height. Compound interest (S3 Unit 7) is a geometric sequence with r = 1 + rate.
Answers
- Is 3, 7, 11, 15, … arithmetic or geometric? Find its 20th term. [2]
Arithmetic with a = 3, d = 4: u₂₀ = 3 + 19 × 4 = 79.
- Find the sum of the first 12 months of savings: 2 000 RWF then 500 RWF more each month. [2]
S₁₂ = 12 ÷ 2 × (2 × 2 000 + 11 × 500) = 6 × 9 500 = 57 000 RWF.
- A ball bounces to 2 m, then each bounce is 0.6 of the one before. How high is the 5th bounce? [2]
u₅ = 2 × 0.6⁴ = 0.259 m.
- What is the sum to infinity of the bounce heights 2 + 1.2 + 0.72 + …? [2]
S∞ = 2 ÷ (1 − 0.6) = 5 m.
- Why does a geometric series with r = 2 have no sum to infinity? [2]
Each term is bigger than the one before, so the sum keeps growing without limit. A sum to infinity exists only when −1 < r < 1.
- Five bacteria double every hour. How many are there after 10 hours (at the start of the 11th hour)? [2]
u₁₁ = 5 × 2¹⁰ = 5 120.
- The 3rd term of an arithmetic sequence is 11 and the 8th term is 31. Find (a) the common difference, (b) the first term, (c) the sum of the first 20 terms. [6]
(a) a + 2d = 11, a + 7d = 31 (1); 5d = 20, d = 4 (1). (b) a = 3 (1). (c) S₂₀ = 10 × (6 + 19 × 4) (2) = 820 (1).
- A geometric sequence has first term 48 and common ratio ½. (a) Find the 6th term. (b) Find the sum of the first 6 terms. (c) Find the sum to infinity. [6]
(a) u₆ = 48 × (½)⁵ = 1.5 (2). (b) S₆ = 48(1 − (½)⁶) ÷ (1 − ½) (1) = 94.5 (1). (c) S∞ = 48 ÷ ½ = 96 (2).
- A ball is dropped from 3 m and each bounce reaches ¾ of the previous height. Find the total distance travelled before it stops. [4]
Bounce heights 2.25, 1.6875, … with r = ¾ (1); sum of bounce heights = 2.25 ÷ ¼ = 9 m (1); each bounce is travelled up and down: 2 × 9 = 18 m (1); total = 3 + 18 = 21 m (1).
- A SACCO pays 5 % interest per year. 100 000 RWF is invested. After how many complete years will it first exceed 150 000 RWF? [4]
100 000 × 1.05ⁿ > 150 000 (1); 1.05ⁿ > 1.5 (1); n > log 1.5 ÷ log 1.05 = 8.3 (1); 9 years (1).