Problem:
If $a_1,a_2,a_3 \cdots a_n$ are in HP then find the value of $ a_1 \cdot a_2 + a_2 \cdot a_3 + a_3 \cdot a_4 + \cdots + a_{n-1} \cdot a_n$
My initial approach,using the property of HP that $ \frac{1}{a_1}, \frac{1}{a_2}, \frac{1}{a_3},\cdots ,\frac{1}{a_n}$ is in AP, I am getting this form:
$$\frac{a_1-a_2}{a_1 \cdot a_2} = \frac{a_2-a_3}{a_2 \cdot a_3} = \cdots = \frac{a_{n-1}-a_n}{a_{n-1} \cdot a_n}= d$$
How to proceed next?