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Finding the nth term of a Quadratic Sequence - GCSE Higher Maths
1st Class Maths
Overview
This video explains how to find the nth term of a quadratic sequence, a sequence where the second difference between consecutive terms is constant. It outlines a step-by-step process: first, calculate the first and second differences to confirm it's quadratic. Then, use half of the constant second difference as the coefficient for the n-squared term. Next, generate the terms for this initial quadratic part and subtract them from the original sequence to reveal a linear sequence. Finally, find the nth term of this linear sequence and combine it with the quadratic part to form the complete nth term of the original quadratic sequence.
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Chapters
- A quadratic sequence has a constant second difference, unlike a linear sequence which has a constant first difference.
- The first difference is found by subtracting consecutive terms.
- The second difference is found by subtracting consecutive terms of the first difference.
Identifying a sequence as quadratic based on its second difference is crucial for knowing which method to apply to find its nth term.
In the sequence 6, 19, 38, 63, 94, the first differences are 13, 19, 25, 31. The second differences are all 6, indicating a quadratic sequence.
- The coefficient of the n-squared term in the nth term formula is always half of the constant second difference.
- Once the n-squared coefficient is found, generate the terms of this partial sequence (e.g., 3n^2).
- This partial sequence represents the quadratic component of the original sequence.
This step isolates the primary quadratic behavior of the sequence, which is essential for building the full nth term formula.
For a second difference of 6, half of it is 3. So, the n-squared term starts as 3n^2. The terms for 3n^2 are 3, 12, 27, 48, 75.
- Subtract the terms of the calculated quadratic part (e.g., 3n^2) from the corresponding terms of the original sequence.
- The resulting sequence will be a linear sequence.
- Find the nth term of this new linear sequence.
By removing the quadratic component, we are left with a simpler linear sequence whose nth term can be easily determined and then added back.
Subtracting the 3n^2 terms (3, 12, 27, 48, 75) from the original sequence (6, 19, 38, 63, 94) yields 3, 7, 11, 15, 19. This is a linear sequence.
- The nth term of the linear sequence found in the previous step is determined by its common difference and starting value.
- Combine the quadratic part (e.g., 3n^2) and the linear part (e.g., 4n - 1) to form the complete nth term.
- The final nth term is the sum of the quadratic and linear components.
This final step synthesizes the identified quadratic and linear components into a single formula that accurately describes any term in the original sequence.
The linear sequence 3, 7, 11, 15, 19 has a common difference of 4 (so it's 4n) and starts at 3 (so it's 4n - 1). Combining this with the 3n^2 part gives the nth term: 3n^2 + 4n - 1.
Key takeaways
- Quadratic sequences are characterized by a constant second difference.
- The coefficient of n^2 in the nth term is half the constant second difference.
- Finding the nth term of a quadratic sequence involves separating it into its quadratic and linear components.
- Subtracting the calculated quadratic part from the original sequence reveals a linear sequence.
- The nth term of the linear sequence is found using standard linear sequence methods (common difference and first term).
- The complete nth term is the sum of the quadratic part and the linear part.
- Mastering linear sequences is a prerequisite for understanding quadratic sequences.
Key terms
Quadratic sequenceLinear sequencenth termFirst differenceSecond differenceConstant differenceCoefficientn-squared term
Test your understanding
- How do you determine if a sequence is quadratic?
- What is the relationship between the second difference and the n-squared term's coefficient?
- What steps are involved in finding the linear part of a quadratic sequence after identifying the quadratic component?
- How do you combine the quadratic and linear parts to get the final nth term formula?
- Why is it important to find the differences between terms in a sequence?