Nth Terms of Quadratic Sequences

# Nth Terms of Quadratic Sequences

GCSE(F), GCSE(H),

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: divide the second difference by 2 to obtain the coefficient of the x^2 term.

To work out the quadratic sequence, determine the an^2 value of each term; then subtract that from the values of the original sequence to obtain a linear sequence. Solve the linear sequence based on bn + c.

## Examples

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

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

Work out the second differences for the first five terms:

Term12 345...
Value1225 4469100...
1st Difference1319 2531...
2nd Difference6 66...

The second difference is 6; the multiple for n^2 is 6 ÷ 2 = 3.

Subtract the value of 3n^2 from the original sequence:

Term12 345...
Original1225 4469100...
3n2312 274875...
Original - 3n2913 172125...
Difference44 44...

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

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`.