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Grade 11 · Quadratic sequences

Quadratic sequences: Linear growth as a special case

Quadratic sequences: investigate linear growth as a special case with quadratic coefficient a = 0; linear coefficient b = 2.

Predict what will happen, press Play, then pause and explain what changed. Every control also works without playback.

Watch the relationship

Paused
Quadratic sequences: Linear growth as a special case. Index n: 0. Term value: 0. Next first difference: 2. Second difference: 0First and second differences0200510sequence valueindex n → · labeled axes rescale to this model
Indices run from zero through ten, and coefficients are real. The displayed next difference may refer to term eleven at the final frame. This formula supplies exact terms, not an inferred rule from a short data sample.

Starts paused. Play once, pause anywhere, or use Step to inspect the mathematics. Playback stops when this panel leaves the screen.

Make it your experiment

Change one value. Notice what follows.

The controls adjust the model. Numbers below describe the current frame. Decimals are rounded.

Index n
0
Term value
0
Next first difference
2
Second difference
0

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Understand what you are seeing

The idea behind the motion.

A quadratic sequence has changing first differences but constant second differences. The linear coefficient changes every first difference by the same amount, while the quadratic coefficient sets the second difference. The case a=0 becomes a linear sequence rather than a quadratic one. This investigation starts with Quadratic coefficient a = 0; Linear coefficient b = 2. Predict the result before playing, then change one parameter while keeping the other fixed to test the reason for the change.

A relationship to keep

uₙ=an²+bn; Δuₙ=a(2n+1)+b; Δ²uₙ=2a

Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.

  1. STEP 1

    Set up the mathematical model

    Read values at whole indices only. The connecting line is not a claim about additional sequence terms. The starting case is “Linear growth as a special case.”

  2. STEP 2

    Follow the changing quantity

    Compare one term with the next using a(2n+1)+b, then compare successive first differences.

  3. STEP 3

    Explain and test the result

    Change the quadratic coefficient's sign to reverse curvature. Set it to zero to distinguish constant first differences from genuine quadratic growth.

Your turn to explain

Make a prediction. Test your reasoning.

Keep Quadratic coefficient a = 0; Linear coefficient b = 2. Pause the timeline at 60%. Given index n = 6, calculate term value, next first difference, second difference. Show the substitution into the displayed formula.

Use the values specified in the question. Reset restores the initial values for this investigation.

Compare your explanation

At n=6, uₙ=(0)(6)²+(2)(6)=12. The next difference is (0)(2(6)+1)+(2)=2, and its next change is 2(0)=0. Results: Term value: 12; Next first difference: 2; Second difference: 0. Decimal values are rounded; retain the original parameters when checking.

Work through a full lesson

Connect the animation to a worked example and practice questions.

Grade 11 · Algebra

Recognize quadratic sequences from their differences

Use a constant second difference to identify a quadratic rule.