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  • 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.

Proof by mathematical induction worked example. Free online a-level further maths core pure 1 notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes.

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


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.