Use second or third differences to identify a polynomial pattern.
Core & Extended · Syllabus C2.7, E2.7
Notes and key methods
A quadratic sequence has constant second differences; a cubic has constant third differences. Compare a simple pattern with \(\displaystyle n^{2}\) or \(\displaystyle n^{3}\), then determine its scale and additive adjustment. The index starts where the question states it.
For \(\displaystyle T_{n}\)=an²+b, the constant second difference is 2a. For \(\displaystyle T_{n}\)=an³+b, the constant third difference is 6a. Verify a proposed rule in all supplied terms; one matching term is insufficient.
Check the conditions
Use the stated units and precision.
Show the method as well as the final answer.
Three worked examples
Example 1 · Simple quadratic and cubic sequences
The sequence 7, 14, 33, 70 has the form \(T_n=An^{3}+B\) with \(\displaystyle n=1,2,3\),… and A>0.
Determine the stated power-pattern rule for \(\displaystyle T_{n}\).
Find the positive integer index at which \(\displaystyle T_{n}=735\).
Write \(\displaystyle T_{n}=An^3+B\). Find B.
Find T at \(\displaystyle n=9\).
Show the reasoning
(a)
The third difference is 6, so the scale is 1. The first term determines the additive constant 6.
Answer: \(T_n=1n^{3}+6\)
(b)
Isolating the power gives \(n^{3}=729\). Taking the positive integer root gives \(\displaystyle n=9\).
Answer: \(9\)
(c)
Compare the supplied powers-of-index pattern, then determine its scale and constant term.
\(6\)
Answer: \(6\).
(d)
Compare the supplied powers-of-index pattern, then determine its scale and constant term.
\(735\)
Answer: \(735\).
Example 2 · Simple quadratic and cubic sequences
The sequence 9, 15, 25, 39 has the form \(T_n=An^{2}+B\) with \(\displaystyle n=1,2,3\),… and A>0.
Determine the stated power-pattern rule for \(\displaystyle T_{n}\).
Find the positive integer index at which \(\displaystyle T_{n}=207\).
Write \(\displaystyle T_{n}=An^2+B\). Find B.
Find T at \(\displaystyle n=10\).
Show the reasoning
(a)
The second difference is 4, so the scale is 2. The first term determines the additive constant 7.
Answer: \(T_n=2n^{2}+7\)
(b)
Isolating the power gives \(n^{2}=100\). Taking the positive integer root gives \(\displaystyle n=10\).
Answer: \(10\)
(c)
Compare the supplied powers-of-index pattern, then determine its scale and constant term.
\(7\)
Answer: \(7\).
(d)
Compare the supplied powers-of-index pattern, then determine its scale and constant term.
\(207\)
Answer: \(207\).
Example 3 · Simple quadratic and cubic sequences
The sequence 11, 32, 89, 200 has the form \(T_n=An^{3}+B\) with \(\displaystyle n=1,2,3\),… and A>0.
Determine the stated power-pattern rule for \(\displaystyle T_{n}\).
Find the positive integer index at which \(\displaystyle T_{n}=656\).
Write \(\displaystyle T_{n}=An^3+B\). Find B.
Find T at \(\displaystyle n=6\).
Show the reasoning
(a)
The third difference is 18, so the scale is 3. The first term determines the additive constant 8.
Answer: \(T_n=3n^{3}+8\)
(b)
Isolating the power gives \(n^{3}=216\). Taking the positive integer root gives \(\displaystyle n=6\).
Answer: \(6\)
(c)
Compare the supplied powers-of-index pattern, then determine its scale and constant term.
\(8\)
Answer: \(8\).
(d)
Compare the supplied powers-of-index pattern, then determine its scale and constant term.
\(656\)
Answer: \(656\).
Common mistakes
Keep intermediate values exact until the final step.