An arithmetic progression has first term \(3\). The sum of the first \(13\) terms of the progression is \(156\).
Find the common difference.[2]
A geometric progression has first term \(3\) and common ratio \(r\). The sum of the first \(13\) terms of the progression is \(156\).
Show that \(r^{13}-52r+51=0\). Show that the common ratio cannot be \(1\) even though \(r=1\) is a root of this equation. Find the possible values of the common ratio.[4]
It is given that the common ratio of the geometric progression is positive, and that the \(n\)th term of this geometric progression is more than \(100\) times the \(n\)th term of the arithmetic progression. Write down an inequality, and hence find the smallest possible value of \(n\).[3]