What happens on Average
In this unit students are introduced to convergence of sequences based on averaging previous terms in the sequence. The power of the “fill-down” function in spreadsheets is exploited. By using various starting numbers, and inferring the limit, students are encouraged to make generalisations of formulae for the convergence of sequences using arbitrary starting numbers.
- infer the precise limit as a fraction from the limit inferred as a decimal number
- set up a spreadsheet to explore the convergence of certain sequences
The convergence of sequences is explored where the rule for constructing series is to average the previous 2 terms in the series using spreadsheets, that is to say
Students make conjectures about the limit of the sequence.
This is extended to averaging the previous 3 terms that is to say
and then this is generalised to the previous k terms.
sequences, convergence, conjectures, proof, weighted average
Ask the students to construct a sequence starting at 1, 9, by averaging the previous two terms. ( So the sequence is 1,9, 5, 7, 6, 6.5...)
- Ask the students to predict where the sequence is heading. They should be able to see that the numbers are getting closer together and that they are approaching a number between 6 and 6.5.
- Given that the manual calculation is tedious assist the students to set up this spreadsheet. (Classes without access to spreadsheets could use calculators but this is tedious).
Notice the need to use the fill down function in columns A and B. Also note the use of the Split screen which shows the sequence value for the 499th term. The second picture displays the formulas needed, though students need never use this mode. Clearly the sequence appears to converge to
- Students can now replace the first two terms by any numbers. Give students a chance to try a few different starting numbers.
- Draw the class together and make a class summary of limits as shown in the table below.
Students need to save their spreadsheet for the next session.
- Discuss as a class what can be stated about the results of their exploration.
What conjectures can you make about this sequence?
What would a graph of the terms of this sequence look like? (You could create the graph easily on Excel)
Does it always converge regardless of what starting terms you use?
What number does it converge to?
You should not expect students to be able to answer this at this stage, but the discussion will be a useful precursor to Session 2.
- Remind students of their work from the previous session. Explain that this session we are going to try to identify a rule. If the first two terms of the sequence are a and b what does the sequence converge to?
- Try a systematic approach. If no students suggest it, suggest that they try keeping a fixed and vary b.
What do you think the limit will be if b is 7? 8? 9? 10?
Now try the same thing with b fixed and varying a.
What do you find?
Is there a pattern?
The students should be able to see that the limit is dependant on both a and b. Guess that the formula is ka + mb, so the task becomes determining the values of k and m.
- Discuss how you could find the values of k and m
- Hopefully the class will suggest that making a = 0 will enable you to solve for b and making b = 0 will enable you to solve for a. The easiest way to do so is to set one variable = 0 and the other = 1.
- On the saved spreadsheet get the students to enter a = 1 and b = 0. It converges to 0.333333333. =
- On the saved spreadsheet enter a = 0 and b = 1. It converges to 0.6666666 =.
So the sequence probably converges to:
- Get the students to confirm this formula works in all cases in the table in session 1. (This is in fact always true for any values of a and b.)
- Copy the saved spreadsheet from the first session and alter it to average the previous three terms rather than two as shown. In the example shown the sequence starts at 1, 9, 3 then the previous are averaged.
- Students explore whether this sequence converges for a variety of initial terms. (It converges for all choices.)
- Discuss “If the first three terms of the sequence are a, b and c what does the sequence converge to?”
- Guess that the formula is ka+mb+nc.
- Discuss what are clever choices of (a, b, c) to find k, m and n – (1,0,0) then (0, 1, 0) then (0,0,1) are smart as they give k, m and n in turn without the need for simultaneous equations.
- On the spreadsheet groups of students explore the convergence using:
(1,0,0) – it converges to 0.166666666 =1/6
(0, 1, 0) – it converges to 0.3333333333 =1/3
(0, 1, 0) – it converges to 0.5 =1/2
So the sequence probably converges to:
Get the students to try some examples to check whether this is so. (This is in fact always true for any a, b, and c.)
- Save the spreadsheet for the next session.
- Gathering the facts from previous sessions:
The sequence a, b, … found by averaging the previous two terms converges to a+2b/3s.
The sequence a, b, c… found by averaging the previous three terms converges to a+2b+3c/6
- Ask students to make conjectures about the convergence of the sequence formed by averaging the previous 4 terms, 5 terms , … k terms.
- Get groups of students to make copies of the previous sessions and modify the spreadsheet. For example averaging the previous five terms starting at 1, 100, 3, 200, 6000 … the spreadsheet would look like this indicating convergence to
- Return together as a class and discuss the findings of individual groups.
What conjectures have been made/proved?
- The following conjecture appears to be correct. Its proof is well beyond the scope of level 5 algebra:
- Let . It converges to the weighted average.
In session 1 the sequence un = 1/2(un-1 + un-2), n≥3 converged for all choices of the first two terms. In this session students explore un = k(un-1 + un-2), n≥3 where k may have any value. Students need to set up a spreadsheet similar to previous ones used and explore convergence for various values of k. This is a challenging task as now there are two dimensions being varied, namely the initial two terms and then the value of k. Students may need help to understand why k should be held constant while varying the starting numbers.
- Assign groups of students different values of k to explore for a range of values of a and b.
- Return together and discuss as a class the findings of each group.
Can you make any conjectures about the results for individual values of k?
Can you make any conjectures about the results for a general value of k?
- The conjectures that should arise are:
If 0<k<1/2 the sequence converges to zero for any choice of the initial 2 values
If k=1/2 the sequence converges to a+2b/3 as found in session 2.
If k>1/2 the sequence diverges for any choice of the initial 2 values.
(These are all correct, though proofs may be beyond the scope Level 5 Algebra)
- Explore un = k(un-1 + un-2 + un-3), n≥ 4 for various values of k.
- Explore un = k(un-1 + un-2 + un-3 + un-4), n≥5 for various values of k.
- Generalise to un = k(un-1 + un-2 + … + un-j), n≥j+1 for various values of k.