Sequences and series: seats in a hall
The school hall has rows of seats: a few at the front and more in each row behind. Change the seats in the first row, the extra seats in each row and the number of rows, and find the seats in the last row and in the whole hall.
Core concept
In an arithmetic sequence the same number d is added each time, so the nth term is a + (n − 1)d, and the sum of n terms is n/2 × (2a + (n − 1)d)
Topic: Sequence and series
A sequence is a list of numbers that follows a rule, like 10, 12, 14, 16, … . When the same number is added each time it is an arithmetic sequence: a is the first term and d the common difference. The second term is a + d, the third a + 2d, so the nth term is tₙ = a + (n − 1)d. Adding the terms gives a series. To add n terms quickly, pair the first with the last, the second with the second last, and so on: each pair adds to the same total, a + tₙ, and there are n/2 pairs. So Sₙ = n/2 × (2a + (n − 1)d).
What you will learn
- Recognise an arithmetic sequence and its common difference.
- Find the nth term with tₙ = a + (n − 1)d.
- Find the sum of n terms with Sₙ = n/2 × (2a + (n − 1)d).
Before you start
- Know how to substitute numbers into a formula.
Materials
- Counters or bottle caps to lay out rows
Equipment
- None
Procedure
- Record a hall with 10 seats in the first row, 2 more in each row, and 10 rows.
- Make it 20 rows and record.
- Make the step 4 seats and record.
- Record 15 seats in the first row, 3 more each row, and 15 rows.
How this simulation is simplified
- Each row has the same number of extra seats as the row in front of it.