3.7: Logarithmic AP and GP with ratio one third

0606/12/F/M/24 — Question 9 · 12 marks

(a)(i)
The first three terms of an arithmetic progression are lg⁡θ2\lg\theta^{2}, lg⁡θ5\lg\theta^{5}, lg⁡θ8\lg\theta^{8}.
Given that the sum to nn terms is 4732lg⁡θ4732\lg\theta, find nn.
[5]
(a)(ii)
This sum equals −14196-14196. Find the exact value of θ\theta.
[1]
(b)(i)
The first three terms of a geometric progression are lg⁡ϕ3\lg\phi^{3}, lg⁡ϕ\lg\phi, lg⁡ϕ1/3\lg\phi^{1/3}.
Determine whether this geometric progression has a sum to infinity.
[2]
(b)(ii)
Find the nnth term in the form lg⁡ϕA\lg\phi^{A}, where AA is a function of nn.
[3]
(b)(iii)
Find ϕ\phi given that the 2020th term is 3−183^{-18}.
[1]