Sequences and series：練習問題

The sum of the first \(n\) terms of an arithmetic series is given by \(S_n = 3n^2 + 2n\). Find the 10th term of this series.

The first three terms of a geometric progression are \(k - 1\), \(2k - 2\), and \(3k\), where \(k\) is a constant. Find the sum of the first 10 terms of this progression.

A convergent geometric series has a first term of 12 and a sum to infinity of 48. Find the value of the 3rd term of the series.

A sequence is defined by the recurrence relation \(a_{n+1} = pa_n + 1\), with \(a_1 = 2\), where \(p\) is a constant. Given that \(\sum_{r=1}^{3} a_r = 18\), find the possible values of \(p\).

In the binomial expansion of \((2 + kx)^6\), where \(k\) is a positive constant, the coefficient of \(x^2\) is 60. Find the coefficient of \(x^3\) in this expansion.

An arithmetic series has first term 5 and common difference 3. If the sum of the first \(n\) terms is 155, find the value of \(n\).

A geometric series has first term 12 and a sum to infinity of 30. Find the smallest value of \(n\) for which the sum of the first \(n\) terms, \(S_n\), exceeds 28.

In the binomial expansion of \((1 + kx)^n\), where \(n\) is a positive integer and \(k\) is a constant, the coefficients of \(x\) and \(x^2\) are 15 and 90 respectively. Find the value of \(n\) and the value of \(k\).

A geometric series has second term 12 and a sum to infinity of 64.

(a) Show that the common ratio, \(r\), of the series satisfies the equation \(16r^2 - 16r + 3 = 0\).
(b) Find the two possible values of \(r\) and the corresponding values of the first term \(a\).
(c) Given that \(r > 0.5\), calculate the sum of the first 5 terms of the series, giving your answer to 2 decimal places.

The first three terms of an arithmetic sequence are \(k, 2k + 3,\) and \(5k - 2\).
(a) Show that \(k = 4\).
(b) Find the common difference \(d\).
(c) The sum of the first \(n\) terms of this sequence is \(S_n\). Find the smallest value of \(n\) such that \(S_n > 2000\).