Teacher page: The sum of an infinite geometric sequence

The one thing to do with the animation

Ask whether a ball that bounces for ever travels an infinite distance.

Most will say yes, because the bouncing genuinely never stops. The total settles on 18 m and the running total is visibly flattening while bounces are still being added.

Hold the slider at 35 or so: the height is four thousandths of a metre and still positive. The process is infinite and the total is not, and that is a genuinely surprising idea worth leaving time for.

The first drop is not doubled

The ball falls 2 m once, and every rebound thereafter is travelled twice. Total = 2 + 2(sum of rebound heights). Students either double everything or double nothing.

And |r| < 1 must be stated. The formula happily returns a number for r = 1.2, and that number is negative, which is impossible for a sum of growing positive terms. Questions are set on exactly this.

The answers

1. a = 10, r = 0.6	10/0.4 = 25.
2. Total distance	2 + 2(1.6/0.2) = 18 m.
3. r = 1.2	B, no sum to infinity exists.

Where the marks go

1 markStating |r| < 1 before applying the formula.

1 markDoubling the rebounds but not the first drop.

1 markAn exact value where one exists.

What each wrong answer tells you

They give	What it means
16.667 (Q1)	Divided by r rather than by 1 − r.
10 (Q2)	Counted each rebound once.
20 (Q2)	Doubled the first drop as well.
8 (Q2)	Gave the sum of the rebound heights without the drop or the doubling.
A (Q3)	Used the formula without checking, and accepted a negative sum of positive terms.

Other things they will say

"Does it ever actually stop?" Mathematically no; physically yes, because the model stops describing reality at very small heights. Worth saying: the model is a model.

"Is 0.999... really 1?" Yes, and this is the cleanest proof they will meet: it is a geometric series with sum 0.9/0.9. Expect argument, and welcome it.

"What if r is negative?" Terms alternate and the sum still exists if |r| < 1. The absolute value in the condition is doing real work.

A possible order

	What is happening
1	Infinite distance? Run the bounces.
2	The formula and its condition.
3	The bouncing ball done properly, with the doubling.
4	Questions where the condition fails.
5	0.999... = 1, and the argument that follows.

Two things not to say

Do not state the formula before the condition. They are one thing.

Do not rush 0.999... = 1. The resistance to it is the most interesting thing in the lesson.

Questions to set

Three tiers, ramping the way practice should: the method on its own, then the method inside something real, then a challenge. Set the tier the class in front of you needs rather than one undifferentiated sheet. Answers are given so these can go straight onto a board.

1Fluency

The method on its own, with friendly numbers. Set these first and move on quickly once they are secure.

Sum to infinityu₁ = 18, r = 1/3. Find S_∞.
27
Test convergenceDoes 5, 7.5, 11.25, ... have a sum to infinity?
No. r = 1.5, so |r| is not less than 1.

2In context

The same skill inside a real situation, where the first job is working out what is being asked.

Bouncing ballA ball dropped from 5 m rebounds to 0.6 of its height. Find the total distance travelled.
5 + 2(3/0.4) = 20 m
Recurring decimalWrite 0.272727... as a fraction using a geometric series.
0.27/(1 − 0.01) = 27/99 = 3/11

3Challenge

Reasoning, working backwards, or spotting an error. These are where the top grades are decided.

Find r from two factsA geometric series has S_∞ = 12 and u₁ = 4. Find r, and find the first n whose total passes 11.9.
r = 2/3. S_n first passes 11.9 at n = 12 (S₁₁ = 11.861, S₁₂ = 11.908).
Reason about the tailFor a series with r = 0.9, the sum to infinity is 10u₁. Explain why 50 terms is still not close, while r = 0.5 needs only about 10.
The shortfall is rⁿ of the total. 0.9⁵⁰ = 0.005 against 0.5¹⁰ = 0.001, and the comparison shows how much slower a near-1 ratio converges.

Practicalities

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