Quadratic sequences

Algebra - Quadratic Sequences

• A quadratic sequence is a sequence of numbers where the differences between terms do not remain constant, but the second differences, i.e., the differences between consecutive differences, do.
• Taking the differences between terms allows you to identify the type of sequence. In a quadratic sequence, the first differences are inconsistent, but the second differences remain constant.

Form of Quadratic Sequences

• The general form of a quadratic sequence is defined as an^2 + bn + c, where a, b, and c are constants and n represents the term number, starting with n = 1 for the first term.
• The second difference gives twice the value of ‘a’ in the general form.

Steps to Find the nth Term Rule for a Quadratic Sequence

• To find the nth term rule of a quadratic sequence, start by finding the first and second differences between consecutive terms.
• Divide the second difference by 2 to find the value of ‘a’.
• Subtract the sequence obtained by the nth term of ‘a’ from your original sequence to get a linear sequence.
• Find the nth term of this linear sequence which will provide ‘b’ and ‘c’ values.
• Combining these values of ‘a’, ‘b’, and ‘c’, gives you the nth term rule.

Key Takeaways

• Practising the process of finding the nth term rule is the most practical way to get better at identifying and working with quadratic sequences.
• Understanding the formula for a generic quadratic sequence (an^2 + bn + c) is pivotal in learning about quadratic sequences.
• Analysis of patterns through differences between terms is central for identifying quadratic sequences from other types of sequences.