Q10 of 36 Page 161

If a1, a2, a3, ..., an are in A.P., where ai > 0 for all i, show that

To prove that:




Multiplying the first term by , the second term by and so on that is rationalizing each term



Using (a + b)(a – b) = a2 – b2



As a1, a2, a3,…,an are in AP let its common difference be d


a2 – a1 = d, a3 – a2 = d … an – an-1 = d


Hence multiplying by -1


a1 – a2 = -d, a2 – a3 = -d … an – an-1 = -d


Put these values in LHS





Multiply divide by



Using (a + b)(a – b) = a2 – b2



The nth term of AP is given by tn = a + (n – 1)d


Where the tn = an is the last nth term and a = a1 is the first term


Hence an = a1 + (n – 1)d


⇒ a1 – an = -(n – 1)d


Substitute a1 – an in LHS




⇒ LHS = RHS


Hence proved


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