Learn with Amar
Teaching video
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Grade 12 chapters and video availability01 · Read and understand
What you will learn
- Use a characteristic equation for a second-order linear recurrence.
- Justify the conclusion "A=B=1, hence aₙ=1+2ⁿ" using the stated assumptions.
Before you start
Sequences and quadratic equations.
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
Solve aₙ₊₂=3aₙ₊₁−2aₙ with a₀=2,a₁=3.
Why this math matters
Use a characteristic equation for a second-order linear recurrence. 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:
- The recurrence has fixed coefficients.
- n is a nonnegative integer.
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.
Solve a recurrence with two independent growth modes
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Solve aₙ₊₂=3aₙ₊₁−2aₙ with a₀=2,a₁=3.
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
Trying aₙ=rⁿ gives r²−3r+2=0
Exponential sequences turn the recurrence into an algebraic equation.
Work through the mathematics
Roots are one and two, so aₙ=A+B2ⁿ
Distinct characteristic roots supply two independent sequence modes.
Check the conclusion
A=B=1, hence aₙ=1+2ⁿ
The two initial conditions determine both constants; a₂=5 matches the recurrence.
The result
A=B=1, hence aₙ=1+2ⁿ
The two initial conditions determine both constants; a₂=5 matches the recurrence.
Common mistakes to catch
- A repeated characteristic root requires a different second solution form.
- Check initial indexing before fitting constants.
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
Find a₄.
Show a hint
Substitute into the closed form.
Reveal answer and explanation
Seventeen
1+16=17.
Practice 2
Why are two initial values needed?
Show a hint
The recurrence looks back two positions.
Reveal answer and explanation
There are two free constants
One starting value cannot generally choose a unique combination of both modes.
Take the idea with you
Compare a recursively generated sequence with a closed formula that exposes its dominant mode.
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: Find all complex roots using equally spaced angles
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