Proof by Induction
Proof by induction is used to prove that a general statement is true for all positive integer values. All proofs by mathematical induction follow four basic steps:
Prove that the general statement is true when n = 1
Assume the general statement is true for n = k
Show that, if it is true for n = 1, the general statement is also true for n = k+1
Conclude that the general statement is true whenever n ∈ ℕ
All four steps must be shown clearly in your workings: prove, assume, show, conclude.
Proving Sums
Often, questions will use the standard results for the sums of r, r² and r³. Regardless, the method for all sums is the same and follows the four steps above.
![Proof by mathematical induction worked example. Free online a-level further maths core pure 1 notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes.](https://static.wixstatic.com/media/744293_bba7ab5b39c24d5f98d55eb96bf21e73~mv2.jpg/v1/fill/w_138,h_179,al_c,q_80,usm_0.66_1.00_0.01,blur_2,enc_auto/744293_bba7ab5b39c24d5f98d55eb96bf21e73~mv2.jpg)
Quick Tip
For the 3rd step, it is generally best to write the last line out first using the general function - just substitute (k+1) into it. You know that this is the answer you want to reach, so use it as a target to help you.
Proving Divisibility Results
Again, follow the four standard steps for proof by induction. For divisibility results, make step 2 equal any general multiple of the divisor:
![Proving divisibility results worked example. Proof by induction. Free online a-level further maths core pure 1 notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes](https://static.wixstatic.com/media/744293_1e22e4c9fdc1415caf9572e226494e24~mv2.jpg/v1/fill/w_138,h_136,al_c,q_80,usm_0.66_1.00_0.01,blur_2,enc_auto/744293_1e22e4c9fdc1415caf9572e226494e24~mv2.jpg)
Proof Using Matrices
Exactly the same four steps apply:
![Proof by induction with matrices, matrix proof by induction. Free online a-level further maths core pure 1 notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes.](https://static.wixstatic.com/media/744293_a70f15e6bd5d4aff846e54a872f8dd67~mv2.jpg/v1/fill/w_138,h_132,al_c,q_80,usm_0.66_1.00_0.01,blur_2,enc_auto/744293_a70f15e6bd5d4aff846e54a872f8dd67~mv2.jpg)