I am stuck on part (b) --> mainly for the proof for n=k+1 step
View attachment 34731
(a)
Using a,b > 0
, this works for a<b and b>a and you can convince yourself if you want. If b>a, both brackets will be negative and that will make a positive, if b<a then both brackets will be positive.
as required.
(b)
:
Hence n = 1 is true as equality holds for the inequality.
Assume k is a positive integer and true for n greater/equal to 1 (or whatever you write for assume stage, up to your discretion)
:
which will be our induction hypothesis
Now proving for
:
We need to prove this inequality;
LHS:
from our induction hypothesis
=
when expanding
=
(1)
We will use
from (i) and substitute this into the eqn 1 to form a new inequality:
(1) <=
(1) <=
and we exactly got RHS of the original statement.
Hence proven by the principles of mathematical induction (insert your own conclusion).