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.
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
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