1. ## Progression problems........

1) In a Geometrical Progression , prove that the product of any two terms, equidistant from the beginning and the end, is constant and is equal to the product of the first and last terms.

2) Prove that the product of the two middle terms of a Geometrical Progression, consisting of an even number of terms is equal to the product of the first and last terms.

let the progression is given by $\displaystyle a, ar, ar^2, ar^3, ....... , ar^{n-1}$ where $\displaystyle r$ is the common ratio and $\displaystyle a$ is the first term and the progression possesses $\displaystyle n$ terms.
let us take $\displaystyle k$th term from the beginning and end, respectively. the terms are then given by $\displaystyle ar^{k-1}$ and $\displaystyle ar^{n-k+1-1}$ (ie, $\displaystyle n-k+1$th term from the beginning). so their product is given by $\displaystyle a^2r^{k-1+n-k+1-1}$ ie $\displaystyle a^2r^{n-1}$.
also, the product of the first and last terms of the progression is $\displaystyle a*ar^{n-1} = a^2r^{n-1}$.