Sequences and series: แบบฝึกหัด

Find the 10th term of an arithmetic sequence whose first term is \(3\) and common difference is \(4\).

An arithmetic series has a fifth term of 18 and a twelfth term of 39. Calculate the sum of the first 20 terms of this series.

A geometric series has a first term \(a\) and a common ratio \(r\). The sum of the first two terms is \(4.8\) and the sum to infinity is \(7.5\). Given that \(r > 0\), find the value of \(r\).

The sum of the first \(n\) terms of an arithmetic series is given by \(S_n = 2n^2 + 3n\). Find the 4th term of the series, \(u_4\).

A geometric series has a first term of 20 and a common ratio of 0.8. Find the smallest value of \( n \) for which the sum of the first \( n \) terms, \( S_n \), satisfies the inequality \( S_n > 92 \).

A geometric series has first term \( a \) and common ratio \( r \). The sum of the first two terms of the series is 15.
The sum to infinity of the same series is 27.

(a) Show that \( r^{2} = \frac{4}{9} \).
(b) Find the two possible values of \( a \).
(c) For the positive value of \( r \), calculate the 5th term of the series, giving your answer to 3 decimal places.
(d) For the negative value of \( r \), find the sum of the first 10 terms of the series, giving your answer to 2 decimal places.