Two positive numbers and are the first and the last terms of a sequence in which…
MATHEMATICS · Multi-part · AoPS · Mathematics · Sequences & Series · Sequences & Series
Problem
Two positive numbers a and b(a=b) are the first and the last terms of a sequence in which exactly n numbers are inserted so that the entire sequence is in geometric progression.
(i) Find an expression for the common ratio of the progression in terms of a, b, and n.
(ii) Write the general term of the inserted numbers and hence obtain the kth inserted number.
(iii) Explain, without numerical computation, why the kth inserted number lies between a and b when a<b.