Can you explain induction and go through an example?

Induction is a method that can be used to prove that a mathematical statement holds for possitive integers, n. It usually consists of four steps, as follows:  1) Basis: Show that the statement to be proved holds for n=1 2) Assumption: Assume that the general statement holds for n=k 3) Inductive: Show that the statement holds for n=k+1 4) Conclusion: Conclude that the statement holds for all possitive integers n Example Given that u= 6 and un+1 = un + 2n + 4 show that un = 2n + 4n It is important to understand that induction must be applied to the statement to be shown (the one we dont know if it holds for all n). In this case this is un = 2n + 4n Basis n=1: u1 = 2+4 = 6 (from general statement). Statement holds for n=1 (equal to given value) n=2: u2 = 4+8 = 12 (from general statement).         u2 = u1 + 2 + 4 = 6 + 2 + 4 = 12 (from requerrence formula)        Statement holds for n=2 Assumption  Assume that the statement holds for n=k  uk = 2k + 4k  Inductive n=k+1 Need to show that: uk+1 = 2k+1 + 4(k+1)  Use requerrence formula:  uk+1​ = uk + 2+ 4 = 2k + 4k +  2 + 4 (using assumption above) uk+1​ = 2k+1 + 4(k+1)     which is equal to the expression that we had to show Therefore, the statement holds for n=k+1 if it holds for n=k Conclusion

The statement holds for n=k+1 if it holds for n=k. Since the stament holds for n=1 and n=2 it now holds for all n, n>=1 by mathematical induction. 

Related Further Mathematics GCSE answers

All answers ▸

How can I find the equation of a straight line on a graph?


Find the solution of 3^{4x} = 9^{(x-1)/2}.


Let Curve C be f(x)=(1/3)(x^2)+8 and line L be y=3x+k where k is a positive constant. Given that L is tangent to C, find the value of k. (6 marks approx)


Find the stationary points of y=x^3 + 3x^2 - 9x - 4


We're here to help

contact us iconContact usWhatsapp logoMessage us on Whatsapptelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo
Cookie Preferences