Prove the following by the PMI :
n7/ 7 + n5 / 5 + n3 / 3 + n2/ 2 - 37/210n is a positive integer for all n belongs to N.
TPT : is a positive integer for all .
to prove this by mathematical induction first we will show that it is true for n = 1
and then we will assume that it is true for n = k and by the help of our assumption we will prove that it is true for n = (k+1)
step I;
put n = 1
which is an integer.
step II: let us assume that result holds for n = k
i.e. is an integer................(1)
step III:
now we will try prove that it is true for n = k+1
all are multiple of 7 . therefore
is a positive integer.
thus ...................(2)
...............(3)
since are multiple of 5. and is a positive integer
......................(4) is a positive integer
from eq(2), (3), (4),(5) and (6):
[from eq(1)]
= sum of positive integers
= a positive integer.
thus we have proved that for n = (k+1); is a positive integer.
thus it is true for all values of n belongs to N.
hope this helps you.
cheers!!