Investigating convergence
The Padovan sequence
1. Continue the Padovan sequence for another 15 terms:
1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16 …
2. Describe in words the rule to find the next term.
3. Calculate the ratio of successive terms, i.e.
1 1 2 2 3 4
, , , , ,
1 1 1 2 2 3
…
4. To what value do these ratios appear to be converging?
5. A more accurate value can be obtained by the ratio of the two terms 128801 and 170625.
The ratio is equal to the real root of the equation 𝑥 3 – 𝑥 – 1 = 0 and its exact value (called
the Plastic Number) is:
3
1
1
23
3
1
1
1
23
√( + √ ) + √( − √ )
2
6
3
2
6
3
or alternatively
1
(9−√69)3 +(9+√69)3
1 2
23 33
Show that these two solutions are equivalent.
6. The sequence can be represented as a geometric spiral of equilateral triangles with sides of
length equal to the terms of the sequence.
Copy and extend this spiral on isometric paper.
The Perrin sequence
This is the Perrin sequence:
3, 0, 2, 3, 2, 5, 5, 7, 10, 12, 17…
1. Find the next ten terms.
2. Find the ratio of successive terms beginning with 3/2, 2/3, 5/2…
3. Why do we not begin the ratios at 0/3?
4. What value do these ratios of terms in the Perrin sequence converge to?
Extension
Investigate how to create the Padovan cuboid spiral.
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Investigating convergence
Teacher notes
Students could use a spreadsheet to investigate the convergence of these sequences.
Padovan sequence:
1. 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, 351, 465, 616, 816,
1081
2. P(n) = P(n - 2) +P(n - 2) with the initial conditions P(0) = P(1) = P(2) = 1
i.e. After 1, 1, 1, term 4 = term 2 + term 1, term 5 = term 3 + term 2 etc.
3. 170625/128801 = 1.324717975792113
4. The plastic number is approximately
1.3247179572447460259609088544780973407344040569
Perrin sequence:
3, 0, 2, 3, 2, 5, 5, 7, 10, 12, 17, 22, 29, 39, 51, 68, 90, 119, 158, 209, 277
Successive terms also converge on the plastic number.
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