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.
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.
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: