A-LevelFurther MathematicsProof by induction9231/12 · Oct/Nov 2016 · Q46 Marks
Using factorials, show that (n/(r-1)) + (n/r) = ((n+1)/r). Hence prove by mathematical induction that (a + x)ⁿ = (n/0)aⁿ + (n/1)aⁿ⁻¹x + ... + (n/r)aⁿ⁻ʳxʳ + ... + (n/n)xⁿ for every positive integer n.
- Using factorials, show that (n/(r-1)) + (n/r) = ((n+1)/r).[2]
- Hence prove by mathematical induction that (a + x)ⁿ = (n/0)aⁿ + (n/1)aⁿ⁻¹x + ... + (n/r)aⁿ⁻ʳxʳ + ... + (n/n)xⁿ for every positive integer n.[4]
Mark scheme
Marking points from the official mark scheme (Oct/Nov 2016). The mark or mark type is shown in brackets.
- [M1] n!/((r-1)!(n-r+1)!) + n!/(r!(n-r)!) = n!/((r-1)!(n-r)!) [1/(n-r+1) + 1/r]
- [A1] = n!/((r-1)!(n-r)!) [r+n-r+1]/(r(n-r+1)) = (n+1)!/(r!(n+1-r)!) = ((n+1)/r)
- [B1] (a+x)¹ = (1/0)a + (1/1)x = a+x ⇒ H₁ is true.
- [B1] Assume Hk is true, i.e. (a+x)ᵏ = (k/0)aᵏ + (k/1)aᵏ⁻¹x + ... + (k/r)aᵏ⁻ʳxʳ + ... + (k/k)xᵏ
- [M1] Multiplying by (a+x), the coefficient of aᵏ⁻ʳ⁺¹xʳ is: ((k)/(r-1)) + ((k)/r)
- [A1] ⇒ Hᵏ⁺¹ is true. Hence Hⁿ is true for all positive integers.
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