Exercises 1
1. Prove that for all natural numbers .
proof)
Let .
Since , is true.
Suppose that is true. i.e.
So, is true.
By mathematical Induction, is true for all natural numbers .
2. Prove for all natural numbers .
proof)
Let .
Since , is true.
Suppose that is true. i.e.
So, is true.
By mathematical Induction, is true for all natural numbers .
3. Prove for all natural numbers .
proof)
Let .
Since , is true.
Suppose that is true. i.e.
So, is true.
By mathematical Induction, is true for all natural numbers .
4. (a) Guess a formula for by evaluating the sum for and . [For , the sum is simply .]
sol.)
So, we guess .
(b) Prove your formula using mathematical induction.
proof)
Let .
Since , is true.
Suppose that is true. i.e.
So, is true.
By mathematical Induction, is true for all natural n
