The nth term in a geometric sequence is given by
For part a you have
Factor out :
Solve the cubic to find the possible values of r.
A geometric progression has first term a where a‡0 and common ratio r where r‡1. the difference between the fourth and first term is equal to four times the difference between the third term and the second term.
i) show that r³ - 4r² + 4r -1 = 0
ii) find two possible values for the ratio of the geometric progression.
ii) for the value of r which the progression is convergent, prove that the sum to infinity is 1/2 a (1 + √5)
The first four terms are: .A geometric progression has first term and common ratio
The difference between is equal to 4 times the difference between
(a) Show that: .
We are told that: .
So we have .
. . Divide by
Factor: .(b) Find two possible values for
We have: .
. . and: .
To be convergent, . Hence, we will use: .(c) For the value of which the progression is convergent,
prove that the sum to infinity is: .
. . Therefore: .