Notes by Keyword

University Engineering

Notes by Category University Engineering

Rate these notesNot a fanNot so goodGoodVery goodBrillRate these notes
  • A-Level Further Maths

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:

  1. Prove that the general statement is true when n = 1

  2. Assume the general statement is true for n = k

  3. Show that, if it is true for n = 1, the general statement is also true for n = k+1

  4. Conclude that the general statement is true whenever 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.

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:

Proof Using Matrices

Exactly the same four steps apply:

White Logo.png

Notes by Category

University Engineering

A-Level Further Maths

Do not download, print, reproduce, or modify these notes in any way. To share, only use the share buttons for each notes sheet or a direct link to a notes sheet or any page on Always credit such sharing with a link that reads ''

If you would like to use these notes in any other way or for any other purpose, please contact us. We will be happy to help accommodate your  requests in a way that protects our work and rights.

© 2021, All Rights Reserved     •     Privacy Policy