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 รท 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`?

Answer: 91

Substitute for n with 10 in the sequence: `10^2 - 10 + 1` = 91`.