A quadratic sequence is given by `U_n=an^2+bn+c`, where `a, b text( and ) c` are constants, `n` is the term and `U_n` is the value of the term. Note that there is no higher power than `n^2` in a quadratic sequence.
The second difference of a quadratic sequence is a constant. Dividing the second difference by 2 gives the coefficient of the `x^2` term.
To work out the quadratic sequence:
work out the first and second differences;
obtain the `a` value by dividing the second difference by 2;
build a sequence using the `an^2` value of each term;
subtract this linear sequence from the original sequence;
work out the `bn + c` from the linear sequence;
add the `an^2` and `bn + c` terms.
What is the `n`th term of the quadratic sequence given by 3, 12, 27, 48, 75, ...?
Work out the second differences for the first five terms:
The second difference is 6; the multiple for `n^2` is 6 ÷ 2 = 3.
This gives a quadratic term of `3n^2`.
Subtract the value of `3n^2` from the original sequence:
|Original - 3n2||9||13||17||21||25||...|
The difference is 4, to give `4n` as that part of the sequence.
Work out the value of the zero term: 9 - 4 = 5. Assemble the parts: `U_n = 3n^2 + 4n + 5`
What is the second term of the sequence `U_n=n^2-n+1`?
Substitute for n with 10 in the sequence: `10^2 - 10 + 1 = 91`.
See also Quadratic Progressions