Sequences & Series — practice questions

Find the general term and the 30th term of the arithmetic sequence $7, 11, 15, 19, \ldots$

A theatre has 18 seats in the front row, and each row behind has 3 more seats than the one in front. If there are 24 rows, how many seats are in the back row, and which row is the first to have more than 60 seats?

In an arithmetic sequence, the 4th term is $p$ and the 9th term is $q$. Find expressions for $d$ and $a$ in terms of $p$ and $q$, and show that the 14th term is $2q - p$.

Find the sum of the first 25 terms of the arithmetic series $6 + 10 + 14 + \cdots$

How many terms of the arithmetic series $5 + 9 + 13 + \cdots$ are needed for the sum to reach 1230?

The sum of the first $n$ terms of an arithmetic series is given by $S_n = 3n^2 + 5n$. Find the first term, the common difference, and a formula for the $n$th term. Explain the relationship between $S_n$ and $u_n$ that you used.

Find the general term and the 9th term of the geometric sequence $5, 15, 45, 135, \ldots$

A ball is dropped from 3 m and rebounds to 65% of its previous height each bounce. Find the height after the 6th bounce, and find the first bounce after which the ball rises less than 20 cm.

The 2nd, 4th and 7th terms of an arithmetic sequence are also, in that order, the 1st, 2nd and 3rd terms of a geometric sequence. Given the arithmetic sequence has first term 4, find its common difference and the common ratio of the geometric sequence.

Find the sum to infinity of the geometric series $24 + 18 + 13.5 + \cdots$

A geometric series has first term 40 and sum to infinity 250. Find the common ratio and the sum of the first 6 terms.

A geometric series has second term 6 and sum to infinity 32. Find all possible values of the first term and the common ratio, and explain why one apparent solution must be rejected.