nth Terms of Quadratic Sequences
nth Terms of Quadratic Sequences
Understanding Quadratic Sequences

A quadratic sequence is a sequence in which the second differences between terms are constant. This signifies that the terms follow a pattern based on a quadratic formula.

To find the nth term of a quadratic sequence, you will need to express the sequence in the form
an^2 + bn + c
.
Determining the Coefficients of the Quadratic Sequence

The coefficient
a
in the nth term is usually half of the second difference in the sequence. 
To find the coefficient
b
, subtractan^2
from each term in the original sequence consequently forming a linear sequence. The common difference in this new sequence will beb
. 
To find the coefficient
c
, substituten = 1
into the expressionan^2 + bn + c
and set it equal to the first term of the sequence. Subsequently, solve forc
.
Application

After finding
a
,b
, andc
, you have the nth term rule for the quadratic sequence. It allows you to find any term in the sequence without listing all the preceding terms. 
Example: If a sequence has a second difference of 4 and starts with 2, 6, 12, the nth term rule is
2n^2  n
.
Considerations
 Always doublecheck your coefficients, as even small mistakes can lead to incorrect results.
 Other sequences such as cubic sequences or Fibonacci sequences require different approaches.