Problem
ALG-B1-M05-P018 A Quadratic Sequence
#18
★★★★★ Level 5 of 5
The sequence \(a_n\) has constant second difference \(6\), with \(a_1=2\), \(a_2=9\). Find \(a_n\).
If the second difference is \(6\), the leading coefficient of the quadratic formula is \(3\).
Let \(a_n=3n^2+bn+c\). From \(a_1=2\): \(3+b+c=2\). From \(a_2=9\): \(12+2b+c=9\).
Subtracting gives \(9+b=7\), so \(b=-2\). Then \(c=1\). Answer: \(a_n=3n^2-2n+1\).
A link between sequences and polynomials.