Mathematics

Problem

Let

(a_n)

be a geometric progression with non–zero terms. (i) Prove that for any positive integers

p

and

q

\dfrac{a_p}{a_q}=r^{\,p-q}

, where

r

is the common ratio. (ii) Hence show that, for any positive integer

k

, one always has

a_{m-k}\,a_{m+k}=a_m^{\,2}

. (iii) Deduce that in any finite geometric progression the product of the first and last terms equals the product of the second and second–last terms, and so on. a. (i) Prove that for any positive integers

p

and

q

\dfrac{a_p}{a_q}=r^{\,p-q}

. b. (ii) Show that for any positive integer

k

a_{m-k}\,a_{m+k}=a_m^{\,2}

. c. (iii) Deduce the corresponding property for the first, last and equidistant terms of a finite geometric progression.

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