My aim with this project is to investigate polynomial sequences, and:
- find a relationship between the degree of the polynomial and the
- number of terms required to find the common difference
- the common difference and the leading coefficient
- find a general 𝑛th term expression for any polynomial sequence of degree 𝑑
In a polynomial sequence 𝑎₃𝑛³ + 𝑎₂𝑛² + 𝑎₁𝑛 + 𝑎₀:
| 𝑛 | 1 | 2 | 3 | 4 | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 𝑢 | 𝑎₃ + 𝑎₂ + 𝑎₁ + 𝑎₀ | 8𝑎₃ + 4𝑎₂ + 2𝑎₁ + 𝑎₀ | 27𝑎₃ + 9𝑎₂ + 3𝑎₁ + 𝑎₀ | 64𝑎₃ + 16𝑎₂ + 4𝑎₁ + 𝑎₀ | ||||||||
| 𝑑₁ | 7𝑎₃ + 3𝑎₂ + 𝑎₁ | 19𝑎₃ + 5𝑎₂ + 𝑎₁ | 37𝑎₃ + 7𝑎₂ + 𝑎₁ | |||||||||
| 𝑑₂ | 12𝑎₃ + 2𝑎₂ | 18𝑎₃ + 2𝑎₂ | ||||||||||
| 𝑑₃ | 6𝑎₃ | |||||||||||
The common 3rd difference shows that the sequence is cubic, and 𝑎₃ is found using the fact that the common 3rd difference is 6𝑎₃. The common difference (2nd and 1st respectively) in quadratic and linear sequences are 2𝑎₂ and 1𝑎₁. In any polynomial sequence of degree 𝑑, this project takes the multiplier of 𝑎𝑑 in the common difference as 𝑞𝑑.
qfind.py shows that for any polynomial sequence of degree 𝑑, 𝑞𝑑 = 𝑑!.