Arithmetic and geometric progressions
Put grains of rice on a row of squares: either add the same number each square (an AP) or multiply by the same number (a GP, like the old chessboard story). Find the last term and the total, and see how fast a GP grows.
Core concept
An AP adds the same d each time: tₙ = a + (n − 1)d, Sₙ = n/2 × (2a + (n − 1)d); a GP multiplies by the same r: tₙ = arⁿ⁻¹, Sₙ = a(rⁿ − 1)/(r − 1)
Topic: Sequence and series
An old story tells of a king who offered a reward on a chessboard: 1 grain of rice on the first square, 2 on the second, 4 on the third, doubling each time. That is a geometric progression (GP): each term is the one before × the same common ratio: A way to compare two amounts, like 2 cups of rice to 3 cups of water (2 : 3). All key words r. Its nth term is tₙ = arⁿ⁻¹ and the sum of n terms is Sₙ = a(rⁿ − 1)/(r − 1). If instead the same number d is added each time, as in 2, 4, 6, 8, …, it is an arithmetic progression (AP): tₙ = a + (n − 1)d and Sₙ = n/2 × (2a + (n − 1)d). An AP grows steadily, by the same amount each step; a GP grows faster and faster. Here the step is d for an AP and r for a GP.
What you will learn
- Tell an arithmetic progression from a geometric progression.
- Find the nth term and the sum of n terms of an AP and a GP.
- Explain why a GP with r > 1 soon becomes much bigger than an AP.
Before you start
- Know number patterns and powers.
Materials
- A handful of rice grains or seeds, and a row of 10 squares drawn on paper
Equipment
- None
Procedure
- Record the AP with a = 2, d = 2 and 5 terms.
- Change to a GP with r = 2 and record.
- Make it 10 terms and record.
- Change back to an AP with 10 terms and record.
How this simulation is simplified
- The step is the common difference d for an AP and the common ratio r for a GP.