Sequences and series: revision Problem set A

The sum of the first n terms of an arithmetic sequence is \(S_n=4n^2-n\).

1. (a) Write down the values of \(S_1\) and \(S_2\).
2. (b) Write down the values of \(u_1\) and \(u_2\).
3. (c) Find d, the common difference.
4. (d) Find \(u_{15}\).
5. (e) Find the greatest value of n for which \(u_n<150\).
6. (f) Find the value of n for which \(S_n=1278\).

The term \(u_5\) of an arithmetic sequence is 40 and the common difference is \(-3\). Find \(u_1\).

An arithmetic sequence has \(u_{6}=17\) and \(u_{14}=49\).

1. (a) Find the common difference.
2. (b) Find the first term.
3. (c) Find \(u_{60}\).
4. (d) Find \(S_{100}\).

The term \(u_4\) of an arithmetic sequence is 20 and the common difference is \(-4\). The first and third terms of this sequence are the first two terms of a geometric sequence. Find the sixth term of the geometric sequence.

The term \(u_2\) of an arithmetic sequence is 22 and the common difference is \(-6\). The first and fourth terms of this sequence are the first two terms of an infinite geometric series. Find the sum to infinity of the geometric series.

A theatre has 20 seats in the first row. Each row after that has 3 more seats than the row in front.

1. (a) Find the number of seats in the 25th row.
2. (b) Find the total number of seats in the first 25 rows.
3. (c) The theatre needs at least 2000 seats. Find the least number of rows it must have.

A factory makes 800 units in its first month. Each month it makes 5% more units than the month before. Give your answers to the nearest whole number.

1. (a) Find the number of units made in month 12.
2. (b) Find the total number of units made in the first 12 months.

A patient takes 60 mg of a drug on day 1. Each later day the dose is 80% of the dose the day before, and this continues for ever. Find the total amount of the drug taken.

An arithmetic sequence has \(S_{10}=215\) and \(S_{20}=830\). Find the first term and the common difference.