Math Problem Statement
Consider the sequence 1,5,13,25,41,...Show that the sequence is quadratic
Solution
To show that a sequence is quadratic, we need to demonstrate that the second differences of the sequence are constant. Quadratic sequences have the form:
Step 1: List the sequence values
We are given the sequence:
Step 2: Find the first differences
The first differences are found by subtracting consecutive terms:
Thus, the first differences are .
Step 3: Find the second differences
Now, we find the second differences by subtracting consecutive first differences:
Since the second differences are constant (), the sequence is quadratic.
Step 4: General form of the quadratic sequence
Given that the second difference is constant and equal to 4, we can conclude that the sequence is generated by a quadratic expression of the form:
Would you like me to find the explicit quadratic formula for this sequence?
Here are 5 follow-up questions to further explore quadratic sequences:
- How do you find the quadratic expression for the sequence given?
- What is the significance of the second differences being constant?
- Can the method used here apply to other sequences? How?
- How would you derive the coefficients , , and of the quadratic sequence?
- Could you use this approach to analyze a cubic sequence?
Tip: When you observe constant second differences, it signals that the sequence follows a quadratic pattern, and you can proceed to derive its equation using standard methods.
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Math Problem Analysis
Mathematical Concepts
Quadratic Sequences
First and Second Differences
Formulas
Quadratic Sequence Formula: an^2 + bn + c
Theorems
Theorem of Constant Second Differences for Quadratic Sequences
Suitable Grade Level
Grades 9-11
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