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:

an=an2+bn+ca_n = an^2 + bn + c

Step 1: List the sequence values

We are given the sequence: 1,5,13,25,41,1, 5, 13, 25, 41, \dots

Step 2: Find the first differences

The first differences are found by subtracting consecutive terms:

51=45 - 1 = 4 135=813 - 5 = 8 2513=1225 - 13 = 12 4125=1641 - 25 = 16

Thus, the first differences are 4,8,12,164, 8, 12, 16.

Step 3: Find the second differences

Now, we find the second differences by subtracting consecutive first differences:

84=48 - 4 = 4 128=412 - 8 = 4 1612=416 - 12 = 4

Since the second differences are constant (44), 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:

an=an2+bn+ca_n = an^2 + bn + c

Would you like me to find the explicit quadratic formula for this sequence?


Here are 5 follow-up questions to further explore quadratic sequences:

  1. How do you find the quadratic expression for the sequence given?
  2. What is the significance of the second differences being constant?
  3. Can the method used here apply to other sequences? How?
  4. How would you derive the coefficients aa, bb, and cc of the quadratic sequence?
  5. 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