**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.**