Mathematics

Problem

In a G.P. with positive first term the following information is given:

T_2-T_5=\dfrac{12}{25}

and

T_3-T_6=\dfrac{12}{125}

, where

T_k

denotes the

k

th term. a. (i) Obtain two equations in

a

and

r

corresponding to the above conditions. b. (ii) Hence, determine the first term

a

and the common ratio

r

, given that

|r|<1

. c. (iii) Find the least positive integer

n

such that the

n

th term of the progression is less than

\dfrac{1}{1000}

. d. (iv) Using your results, calculate the sum to infinity of the progression.