Modelling Geometric Series | CAIE 9709 Pure Mathematics 1 Notes

Pure Mathematics 1

Mathematics9709

Turn a recurring decimal into a GP

Find the repeated contribution before choosing a sum formula. In $0.272727\ldots$ , the two-digit block 27 repeats two decimal places later each time:

0.272727\ldots=\frac{27}{100}+\frac{27}{10000}+\cdots

The first term is $27/100$ and the ratio is $1/100$ . Its sum is $(27/100)/(1-1/100)=3/11$ . This is exact because the decimal continues forever.

Keep any non-repeating part separate. For example, $0.1272727\ldots$ is $0.1$ plus a GP starting at $27/1000$ , still with ratio $1/100$ .

Count each journey, not just each height

A ball is dropped from 8 m. Each rebound reaches half the previous height. The rebound heights are 4, 2, 1, … metres, a GP. Total distance also includes the downward journeys.

Ground impacts

Journey	Distance
Initial drop	8 m
First rebound	4 + 4 = 8 m
Second rebound	2 + 2 = 4 m

Total: 8 + 8 + 4 = 20 m

Heights are to scale. Journeys are separated horizontally for reading.

The initial 8 m is travelled once. Each complete rebound height is travelled twice: once upwards and once downwards.

D_\infty=8+2(4+2+1+\cdots)

D_\infty=8+2\left(\frac4{1-1/2}\right)=24\text{ m}

To the third ground impact, there are only two complete rebounds:

D=8+2(4+2)=20\text{ m}

If the journey ends at the top of a rebound, count that final upward journey once. The GP assumes the same height ratio at every bounce; it is a model, not a claim about every real ball.

Now you try

Your turnIndependent practice

(a) Write $0.16666\ldots$ as an exact fraction using a series.

(b) A ball is dropped from 10 m. Each rebound reaches 60% of its previous height. Find the total distance to the third impact, and the limiting total distance.

Show worked answer

(a) Separate $1/10$ from the repeating part:

\frac1{10}+\frac{6/100}{1-1/10}=\frac16

(b) The first two rebound heights are 6 m and 3.6 m.

D_3=10+2(6+3.6)=29.2\text{ m}

D_\infty=10+\frac{2(6)}{1-0.6}=40\text{ m}