Exercise XXVII. Problem 30.
If a, b, c are in AP, prove that ,
,
are in AP.
Proof:
Given that
TPT: . —— Equation 1
Let us try to utilize the following formulae:
which implies the following:
and
Our strategy will be reduce LHS and RHS of Equation I to a common expression/value.
which is equal to
which is equal to
which is equal to
which in turn equals
From the above, consider only the expression, given below. We will see what it simplifies to:
—- Equation II.
Now, consider RHS of Equation I. Let us see if it also boils down to the above expression after simplification.
From equation II and above, what we want is given below:
that is, want to prove that
but, it is given that and hence,
, which means
and
that is, want to prove that
i.e., want:
i.e., want:
i.e., want:
Now, in the above,
.
Hence, .
QED.