Sequences and series：練習問題

The sum of the first \(n\) terms of a sequence is given by \(S_n = 3n^2 - 5n\). Find an expression for the \(n\)-th term, \(u_n\), for \(n \ge 1\).

In an infinite geometric progression, the sum to infinity is 6 and the sum of all the even-indexed terms (the 2nd, 4th, 6th terms, and so on) is 2. Find the first term \( a \) and the common ratio \( r \) of the progression.

A sequence \(u_1, u_2, u_3, \dots\) is defined by \(u_1 = 3\) and the recurrence relation \(u_{n+1} = \frac{u_n}{2u_n + 1}\) for all integers \(n \ge 1\).
Find the exact value of \(\sum_{n=1}^{50} \frac{1}{u_n}\).

The first term of a geometric progression is \(24\) and the common ratio is \(-\frac{1}{3}\). Find the exact sum to infinity, \(S_\infty\), of the progression.

An arithmetic progression has first term \(a\) and common difference \(d\), where \(d \ne 0\). The first, fifth, and seventeenth terms of this arithmetic progression are the first three consecutive terms of a geometric progression. Find the common ratio of the geometric progression.

A sequence \(u_1, u_2, u_3, \dots\) is defined by the recurrence relation \(u_{n+1} = \frac{1}{2}u_n + 3\) for \(n \ge 1\), with \(u_1 = a\). Determine the set of values of \(a\) such that the sequence is strictly decreasing. Hence, find the sum to infinity of the series \(\sum_{r=1}^\infty (u_{r+1} - u_r)\) in terms of \(a\).

The sum of the first \( n \) terms of a series is given by \( S_n = 3n^2 - 2n \). Show that the sequence of terms forms an arithmetic progression, and find the least value of \( n \) such that \( S_n \) exceeds 500.

The first term of an arithmetic progression (AP) is \( a \) and the common difference is \( d \), where \( a, d \neq 0 \). The first, fourth, and twelfth terms of the AP are the first three terms of a geometric progression (GP) with common ratio \( r \).
(i) Show that \( r = \frac{8}{3} \).
(ii) Given that \( a = 2 \), find the sum of the first \( n \) terms of the AP, \( S_n \), in terms of \( n \).
(iii) Find the least value of \( n \) such that the sum of the first \( n \) terms of the GP exceeds \( 10^6 \).