Series：練習問題

The first three terms of an arithmetic progression are 5, 11, and 17. Find the 20th term of this progression.

The sum of the first \(n\) terms of an arithmetic progression is given by \(S_n = 2n^2 + 5n\). Find the 10th term of the progression.

A geometric progression has a first term of 5 and a common ratio of 1.2. Find the smallest value of \(n\) such that the sum of the first \(n\) terms exceeds 100.

Find the coefficient of \(x^3\) in the expansion of \((2 + 3x)^5\).

The first two terms of a geometric progression are 12 and 9. Find the sum to infinity, \(S_{\infty}\), of this progression.

An arithmetic progression has first term \(a\) and common difference \(d\). The sum of the first 10 terms is 210. Given that the 2nd, 4th, and 8th terms of this arithmetic progression are the first three terms of a geometric progression, find the value of \(a\) and the value of \(d\).

A geometric progression has first term \(a\) and common ratio \(r\). An arithmetic progression has first term \(a\) and common difference \(d\).
The 1st, 2nd and 3rd terms of the geometric progression are equal to the 1st, 3rd and 5th terms of the arithmetic progression respectively.
(a) Express \(d\) in terms of \(a\) and \(r\).
(b) Show that \(r = 1\) is a possible value for the common ratio and find the other possible value.
(c) If \(r\) takes the value not equal to 1, find the sum of the first 10 terms of the arithmetic progression in terms of \(a\).