The recurrence relation for
which arises in the calculation of the efficiency of merge sort, is
Show that for every integer ,
For , where the recurrence relation for is given by
Recurrence relation for is,
To prove a statement there are three steps in mathematical induction,
Step 1: Prove that the given statement is true for .
Step 2: Let the given statement is true for , where k is some positive integer greater than 1.
Step 3: In this step, prove that the given statement is true for .
Given recurrence relation for is,
Consider, the statement as:
Put in equation (1),
So, the statement is true for .
Let the statement is true for , so from equation (1),
Now put in equation (1),
Consider left side of equation (2),
For , where the recurrence relation for , is given by
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