How to decode the hidden patterns of quadratic sequences
In short: Move beyond simple addition to understand sequences where the rate of change is itself changing. This video demonstrates a systematic method for deriving the nth term formula of any quadratic sequence.
A sequence is classified as quadratic if its second difference remains constant. Unlike linear sequences, where the first difference is steady, quadratic sequences require looking at the change between those differences. For example, in the sequence 5, 9, 15, 23, 33, the first differences are 4, 6, 8, and 10, while the second difference is a constant 2.
To find the nth term rule, you first halve the constant second difference to determine the coefficient of the n-squared term. By subtracting this n-squared sequence from the original, you can identify the remaining linear pattern. In the case of 5, 12, 23, 38, 57, halving the second difference of 4 gives a 2n-squared term, leaving a linear remainder of n + 2 to form the final formula: 2n² + n + 2.
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Sources
- Summary by Intellect · source: Quadratic Sequences: nth Term | Algebra | Maths | FuseSchool
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