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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?

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<i>Harmonic progressions</i> are formed by taking the reciprocals of arithmetic progressions. –  Yuval Filmus Dec 5 '10 at 20:22
    
$a_i\cdot a_{i+1}=\frac{a_i-a_{i+1}}{d}$. So all terms in your series have the same denominator. What to you get when you add the numerators? –  Timothy Wagner Dec 5 '10 at 20:33
    
Timothy Wagner: It is Solved now ... :-) –  Quixotic Dec 5 '10 at 20:44

1 Answer 1

up vote 2 down vote accepted

Write $a_i a_{i+1}$ as a difference of two fractions to get a telescopic sum.

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