Sequences and series: exam review Exam review

An arithmetic sequence has \(u_{3}=19\) and \(u_{10}=47\). [5]

1. (a) Find the common difference. [2]
2. (b) Find the first term. [1]
3. (c) Find the sum of the first 20 terms. [2]

A library lends 1200 books in its first month. In each later month it lends 4% more books than in the month before. [6]

1. (a) Find the number of books lent in month 9. Give your answer to the nearest whole number. [2]
2. (b) Find the total number of books lent in the first 12 months. Give your answer to the nearest whole number. [2]
3. (c) Find the first month in which more than 2000 books are lent. [2]

The first three terms of a geometric series are \(40+30+22.5+\dots\) [5]

1. (a) Write down the common ratio. [1]
2. (b) Find the 8th term. [2]
3. (c) Find the sum to infinity. [2]

Two questions on infinite geometric series. [5]

1. (a) Write \(0.\overline{36}\) as a fraction in lowest terms. [3]
2. (b) The series \((x-3)+(x-3)^2+(x-3)^3+\dots\) converges. Find the range of values of \(x\). [2]

The sum of the first \(n\) terms of an arithmetic sequence is \(S_n=3n^2+5n\). [13]

1. (a) Find \(S_1\) and \(S_2\). [2]
2. (b) Hence write down \(u_1\) and \(u_2\), and find the common difference. [2]
3. (c) Find \(u_{20}\). [2]
4. (d) Find the greatest value of \(n\) for which \(u_n<150\). [2]
5. (e) Find the value of \(n\) for which \(S_n=1300\). [2]
6. (f) The first and third terms of this arithmetic sequence are the first two terms of a geometric sequence. Find the sixth term of the geometric sequence. [3]

A ball is dropped from a height of 10 m onto a hard floor. After each bounce it rises to 75% of the height of the bounce before. [12]

1. (a) Find the height the ball reaches after the 5th bounce. [2]
2. (b) Find the number of bounces after which the ball first rises to less than 0.5 m. [3]
3. (c) Show that the total distance the ball has travelled when it hits the floor for the third time is 36.25 m. [3]
4. (d) Find the total distance the ball travels before it comes to rest. [4]

A geometric series has second term 12 and sum to infinity 75. [12]

1. (a) Show that \(25r^2-25r+4=0\). [2]
2. (b) Find the two possible values of the common ratio. [2]
3. (c) Find the corresponding values of the first term. [2]
4. (d) For the series with the larger common ratio, find the sum of the first 10 terms. [3]
5. (e) For this series, find the least value of \(n\) for which \(S_n>74\). [3]