The Tribonacci Numbers are defined by To = T₁ = 0, T₂ = 1, and Tn = Tn-1 + Tn-2 +Tn_3. To derive an analogue of the Binet formula, we assume the tribonacci formula is of the form ● ● ● Tn = A + Bon + Co2. #1. Calculate Tn up to n = 20. Then calculate the Tn ratios Let to be the 20th ratio with six decimal Tn-1 places. #2. In the graph of f(x) = x³ - x² - x - 1 at right, the single real zero is approximately do. Verify by evaluating f(x) at po. #3. (Optional) Use synthetic division to factor f(x) = (x - ₂)q(x).
The Tribonacci Numbers are defined by To = T₁ = 0, T₂ = 1, and Tn = Tn-1 + Tn-2 +Tn_3. To derive an analogue of the Binet formula, we assume the tribonacci formula is of the form ● ● ● Tn = A + Bon + Co2. #1. Calculate Tn up to n = 20. Then calculate the Tn ratios Let to be the 20th ratio with six decimal Tn-1 places. #2. In the graph of f(x) = x³ - x² - x - 1 at right, the single real zero is approximately do. Verify by evaluating f(x) at po. #3. (Optional) Use synthetic division to factor f(x) = (x - ₂)q(x).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please explain as well. Thank you!
Transcribed Image Text:The Tribonacci Numbers are defined by To = T₁ = 0, T₂ = 1, and
0
Tn = Tn-1 + Tn_2 + Tn_3.
To derive an analogue of the Binet formula, we assume the
tribonacci formula is of the form
Tn = Apő + Bor + Co2.
#1. Calculate Tn up to n = 20. Then calculate the
ratios_™n. Let o be the 20th ratio with six decimal
Tn-1
places.
2
#2. In the graph of f(x) = x³ - x² - x - 1 at right,
the single real zero is approximately do. Verify by
evaluating f(x) at po.
2
#3. (Optional) Use synthetic division to factor
f(x) = (x - ₁) g(x).
#4. (Optional)
Use the quadratic formula on q (x) to approximate the other (complex
roots) ₁ and 2. Find the lengths |0₁| and 1₂1. In the formula, these numbers tend
to zero for large n.
#5. The value of Tn+1 is obtained by multiplying Tn by do and rounding to the nearest
integer. Calculate up to T20 this way, starting at T15, and compare with the values from
the recursive rule.
#6. Use your data set to estimate A in the approximation T~Aon, approximate T21,
and compare with the actual value.
#7 (Optional) Program calculations where appropriate. For ranges of values, put output
in table form. Print and submit code also.
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Transcribed Image Text:The Tribonacci Numbers are defined by To = T₁ = 0, T₂ = 1, and
0
Tn = Tn-1 + Tn_2 + Tn_3.
To derive an analogue of the Binet formula, we assume the
tribonacci formula is of the form
Tn = Apő + Bor + Co2.
#1. Calculate Tn up to n = 20. Then calculate the
ratios_™n. Let o be the 20th ratio with six decimal
Tn-1
places.
2
#2. In the graph of f(x) = x³ - x² - x - 1 at right,
the single real zero is approximately do. Verify by
evaluating f(x) at po.
2
#3. (Optional) Use synthetic division to factor
f(x) = (x - ₁) g(x).
#4. (Optional)
Use the quadratic formula on q (x) to approximate the other (complex
roots) ₁ and 2. Find the lengths |0₁| and 1₂1. In the formula, these numbers tend
to zero for large n.
#5. The value of Tn+1 is obtained by multiplying Tn by do and rounding to the nearest
integer. Calculate up to T20 this way, starting at T15, and compare with the values from
the recursive rule.
#6. Use your data set to estimate A in the approximation T~Aon, approximate T21,
and compare with the actual value.
#7 (Optional) Program calculations where appropriate. For ranges of values, put output
in table form. Print and submit code also.
Solution
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