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● 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 = Apő + Bo1 + Co2. #1. Calculate Tn up to n = 20. Then calculate the ratios. Let o 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-po) q(x). #4. (Optional) Use the quadratic formula on q (x) to approximate the other (complex roots) ₁ and 2. Find the lengths |p₁| and ₂1. 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~Ad, 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.

● 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 = Apő + Bo1 + Co2. #1. Calculate Tn up to n = 20. Then calculate the ratios. Let o 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-po) q(x). #4. (Optional) Use the quadratic formula on q (x) to approximate the other (complex roots) ₁ and 2. Find the lengths |p₁| and ₂1. 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~Ad, 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.

Advanced Engineering Mathematics
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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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.
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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