Lesson 6 of 10 · 7 min
Proof by induction
NCERT §7.2.1
A visitor at the stall asks: the pattern works for small powers, but how do we know it works for every power, even 100? Checking cases one by one can never finish.
The lesson in notes
In short
The theorem is proved by the principle of mathematical induction. Let P(n) be the statement (a + b)ⁿ = nC0 aⁿ + nC1 aⁿ⁻¹b + … + nCn bⁿ.
Base step: for n = 1 the right side is 1C0 a + 1C1 b = a + b, so P(1) is true.
Inductive step: assume P(k) is true for some positive integer k. Then (a + b)ᵏ⁺¹ = (a + b)(a + b)ᵏ, and the bracket for (a + b)ᵏ is known from P(k).
Multiplying by a raises every power of a by 1; multiplying by b raises every power of b by 1. The two resulting rows are the same coefficients shifted one place, and like terms are then collected.
The collected coefficient of aᵏ⁺¹⁻ʳbʳ is kCr + kC(r − 1), which equals (k + 1)Cr by Pascal's rule. The end coefficients stay 1 because kC0 = 1 = (k + 1)C0 and kCk = 1 = (k + 1)C(k + 1). So P(k + 1) is true.
Since P(1) holds and P(k) always leads to P(k + 1), P(n) is true for every positive integer n.