In a convergent geometric progression, the sum of the first $n$ terms is equal to the sum of the r

Question

In a convergent geometric progression, the sum of the first $n$ terms is equal to the sum of the remaining terms. Given also that the $(n+1)$ th term is $\frac{1}{2}$, determine the value of the $(2n+1)$th term.
[4]
An arithmetic progression has first term $a$ and common difference $d$. The eighth, third and second term of the progression are successive terms of an infinite geometric progression.
If the first term of the geometric progression is $10$, find the sum of the even-numbered terms of the progression.[4]
A sequence is formed in which the $n$ th term is given by $\ln \left| {{u}_{n}}{{u}_{n+1}} \right|$, where ${{u}_{n}}$ is the $n$ th of the geometric progression.
Show that the sequence forms an arithmetic progression and state the value of the common difference.[3]

Accepted Answer

(a) ${\frac{1}{4}}$ (b) ${\frac{25}{12}}$ (c) ${-2\ln 5}$

2009 NYJC P1 Q9