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. 

AS
Answered by Adamos S. Further Mathematics tutor

2486 Views

See similar Further Mathematics GCSE tutors

Related Further Mathematics GCSE answers

All answers ▸

y = (x+4)(6x-7). By differentiating, find the x coordinate of the maximum of this equation.


Consider the Matrix M (below). Find the determiannt of the matrix M by using; (a) cofactor expansion along the first row, (b) cofactor expansion along the second column


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)


How do you use derivatives to categorise stationary points?


We're here to help

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

© MyTutorWeb Ltd 2013–2025

Terms & Conditions|Privacy Policy
Cookie Preferences