## Nth Terms of Quadratic Sequences

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.

## Example 1

What is the nth term of the quadratic sequence given by 3, 12, 27, 48, 75, ...?

Work out the second differences for the first five terms:

 Term 1 2 3 4 5 ... Value 12 25 44 69 100 ... 1st Difference 13 19 25 31 ... 2nd Difference 6 6 6 ...

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:

 Term 1 2 3 4 5 ... Original 12 25 44 69 100 ... 3n2 3 12 27 48 75 ... Original - 3n2 9 13 17 21 25 ... Difference 4 4 4 4 ...

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

Answer: U_n=3n^2+4n+5

## Example 2

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.