Q2 of 66 Page 12

Prove the following by the principle of mathematical induction:


To prove: Prove that by the Mathematical Induction.

Let Assume P(n):12 + 22 + 32 + - - - + n2 =


For n = 1


P(1): 1 =


1=1


= P(n) is true for n = 1


Let P(n) is true for n = k, so


P(k): 12 + 22 + 32 + - - - - - + k2 =


Let’s check for P(n) = k + 1, So


P(k): 12 + 22 + 32 + - - - - - + k2 + (k + 1)2 =


= 12 + 22 + 32 + - - - - - + k2 + (k + 1)2


=


=


=


=


=


=


=


Therefore, P(n) is true for n = k + 1


Hence, P(n) is true for all n∈N by PMI


More from this chapter

All 66 →
6

If P(n) is the statement “n2 – n + 41 is prime”, prove that P(1), P(2) and P(3) are true. Prove also that P(41) is not true.

Given. P(n) = n2 - n + 41 is prime


Prove. P(1),P(2) and P(3) are true and P(41) is not true.

 1

Prove the following by the principle of mathematical induction:

i.e., the sum of the first n natural numbers is

3

Prove the following by the principle of mathematical induction:

 4

Prove the following by the principle of mathematical induction: