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Nth Terms of Quadratic Sequences

Nth Terms of Quadratic Sequences

A quadratic sequence is given by Un=an2+bn+c, where a,b and c are constants, n is the term and Un is the value of the term. Note that there is no higher power than n2 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 x2 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 an2 value of each term;

subtract this linear sequence from the original sequence;

work out the bn+c from the linear sequence;

add the an2 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 n2 is 6 รท 2 = 3.

This gives a quadratic term of 3n2.

Subtract the value of 3n2 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: Un=3n2+4n+5

Answer: Un=3n2+4n+5

Example 2

What is the second term of the sequence Un=n2-n+1?

Substitute for n with 10 in the sequence: 102-10+1=91.

Answer: 91

See also Quadratic Progressions