Binomial theorem, Pascal's triangle, and proof by induction
MathematicsAlgebraAges 15–16
Use it with your class: students open it from a link or a QR code, with no account needed.
Next in Algebra
The expansion of (a + b)ⁿ has coefficients equal to row n of Pascal's triangle. Increase n step by step to see how each new row is generated from the previous one via the identity Cₙᵏ⁻¹ + Cₙᵏ = Cₙ₊₁ᵏ – exactly the induction step P(n) ⇒ P(n + 1) – alongside a falling-domino chain illustrating the principle of induction, the general term, the coefficient sum 2ⁿ, and the largest term once a and b are set to numbers.
Lesson: Mathematical induction, the binomial theorem
What it shows
The binomial theorem says that (a + b)ⁿ is the sum of the terms Cₙᵏaⁿ⁻ᵏbᵏ for k = 0 to n, so its n + 1 coefficients form row n of Pascal's triangle. Each row comes from the previous one by Pascal's rule, Cₙᵏ⁻¹ + Cₙᵏ = Cₙ₊₁ᵏ, which is exactly the inductive step P(n) ⇒ P(n + 1) when the expansion is multiplied by (a + b). Falling dominoes model proof by induction: a true base case plus a valid inductive step makes every case true. Setting a = b = 1 gives the coefficient sum 2ⁿ.
How to use
Increase Exponent n one step at a time and watch each new row of Pascal's triangle form from the two cells above each entry. Click a cell, or set Term k + 1 to inspect, to highlight one term of the expansion. Then enter numbers in Value of a and Value of b to see every term's value, the total (a + b)ⁿ and the term with the largest absolute value. Try values of opposite sign to see the terms alternate.
Parameters you can change
- Exponent n 1–12
- Term k + 1 to inspect 0–12
- Value of a -5–5
- Value of b -5–5
Questions to explore
- How many terms does the expansion of (a + b)⁵ have, and what are its coefficients?
- Why does the sum of the coefficients in row n equal 2ⁿ?
- In the domino chain, what corresponds to the base case, and what goes wrong with a proof that skips it?
