Question

(a) use mathematical induction to show that 1 + 3 +.....+(2n + 1) = (n + 1)^2 for all n e N,n>1.(b) n<2^n for all n,n is greater or equels to 1

Answer 1

To prove by Mathematical Induction:

                           (1)

Proof:
Step 1:

for n = 1:
LHS = 1 + 3 = 4

RHS = (1 + 1)² = 4

So, (1) is satisfied for n = 1

Step 2:
Assume the formula to be true for n = k.

i.e.,

Step 3:

To prove that (1) is true for n = k+1 also.

i.e.,

To prove:

                         (2)

Step 4:

LHS of (2) is:

= RHS of equation (2)

Thus, it is proved that the given relatio (1) is true for n = k +1 if it true for n = k.

But, we have proved that it is true for n = 1.

So, by mathematical induction, it is true for all n, for n 1