Counting Pills

An Investigation

Counting pillsPharmacists sometimes use a triangular tray to quickly count

pills. The pills are poured into the tray and fill up the tray level by level. The numbers written on the tray will be used to help

quickly count the number of pills in the tray.

The diagram shows the case when 15 pills are resting on the tray.

15 pills

21 pills28 pills

How could the tray be used to measure out

a) 22 pills b) 34 pills

c) 60 pills d) 200 pills

Imagine if the tray was larger.

The 5th number on the try is 15.

The 6th number is 21.

The 7th is 28.

Copy and complete

the table up to the

fifteenth number

Number written on the tray

Pills in the tray

1st 1

2nd 3

3rd 6

4th 10

5th

6th

7th

8th

9th etc

15 pills

21 pills28 pills

By forming a difference table, or otherwise, show that the nth number on the tray is given by

Use this formula to find the 40th number written on the tray.

How many numbers will have to be written on the tray if you want to use the tray to count out 7021 pills?

)1(2

1nn

SolutionThe difference table has second differences constant

so a quadratic formula will fit the data.

This quadratic turns out to be

Using this formula then the 40th number to be written on the tray will be

This means that 820 pills will be able to be counted up to the 40th number.

)1(2

1nn

820)41(402

1

If you wish to count up to 7021 pills then you require n such that

Cross multiplying and simplifying

gives n2 + n – 14042 = 0 and this factorizes

as ( n – 118)(n+ 119) = 0, giving n = 118.

7021)1(2

1nn