Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Grade 11 chapters and video availability01 · Read and understand
What you will learn
- Use a constant second difference to identify a quadratic rule.
- Justify the conclusion "b=0,c=1, so aₙ=n²+1" using the stated assumptions.
Before you start
Sequences and polynomial evaluation.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
The sequence begins 2,5,10,17,26 for n=1,…,5. Find a quadratic formula.
Why this math matters
Use a constant second difference to identify a quadratic rule. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- A quadratic rule in n is assumed.
- Indices are consecutive and equally spaced.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Recognize quadratic sequences from their differences
PausedQuestion: Start with the question. Paused.
Question
Start with the question
The sequence begins 2,5,10,17,26 for n=1,…,5. Find a quadratic formula.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
First differences are 3,5,7,9; second differences are 2,2,2
Equal-step quadratic sequences have constant second difference 2a.
Work through the mathematics
For an²+bn+c, a=1; the first two values give b+c=1 and 2b+c=1
Matching values determines the remaining coefficients.
Check the conclusion
b=0,c=1, so aₙ=n²+1
Checking n=5 gives 26 and confirms the displayed pattern.
The result
b=0,c=1, so aₙ=n²+1
Checking n=5 gives 26 and confirms the displayed pattern.
Common mistakes to catch
- Constant first difference characterizes a different, linear pattern.
- Finite data alone do not prove a unique continuation.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
What is the sixth predicted term?
Show a hint
Substitute n=6.
Reveal answer and explanation
37
36+1=37.
Practice 2
Does a finite list uniquely prove an infinite rule?
Show a hint
Other formulas can agree at finitely many indices.
Reveal answer and explanation
No
The quadratic-model assumption is needed for this extrapolation.
Take the idea with you
State a model assumption before extrapolating a numerical pattern.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Predict whether a polynomial crosses or touches an axis
Completion is your own study record, not a test score. It stays in this browser, does not sync to another device, and can be removed by clearing browser data.