Question

In a finite GP, prove that the product of the terms equidistant from the beginning and end is the product of first and last terms.

Answer 1

We need to prove that the product of the terms equidistant from the beginning and end is the product of first and last terms in a finite GP. 

Let us first consider a finite GP.

A, AR, AR²….AR^{n -1} , ARⁿ . 

Where n is finite. Product of first and last terms in the given GP = A.ARⁿ = A²Rⁿ → (a) 

Now, n^th term of the GP from the beginning = AR^n-1 → (1) 

Now, starting from the end, 

First term = ARⁿ 

Last term = A

\(\frac{1}{R} \) = Common Ratio

So, an n^th term from the end of GP, A_n = (ARⁿ)\(\big(\frac{1}{R^{n-1}} \big)\) = AR → (2)

So, the product of nth terms from the beginning and end of the considered GP from (1) and (2) = (AR^n-1 ) (AR)

= A²Rⁿ → (b) 

So, from (a) and (b) its proved that the product of the terms equidistant from the beginning and end is the product of first and last terms in a finite GP.