The ratio of the sum of the second and thrid terms of a geometric progression to its fourth term is 3:4. Show that 3r^2-4r-4=0,where r is common ratio

Question

If r>0 and the fifth term of this progression is 48, find its tenth term.

Answer 1

The sequence has the form a, ar, ar², ar³, ...

(ar + ar²) / ar³ = 3/4

4(ar + ar²) = 3ar³

In a geometric sequence, a≠0 and r≠0.

So, we can divide by ar to get 4 + 4r = 3r².

Thus, 3r² - 4r - 4 = 0.

So, (3r + 2)(r - 2) = 0

r = -2/3 or 2

Since r>0, we can throw out -2/3.

Fifth term = ar⁴ = 16a = 48. So, a = 3.

Tenth term = ar⁹ = 3(2)⁹ = 1536