Convergence conditions

7

1. The first two terms of an arithmetic progression are 1 and \(\cos ^ { 2 } x\) respectively. Show that the sum of the first ten terms can be expressed in the form \(a - b \sin ^ { 2 } x\), where \(a\) and \(b\) are constants to be found.
2. The first two terms of a geometric progression are 1 and \(\frac { 1 } { 3 } \tan ^ { 2 } \theta\) respectively, where \(0 < \theta < \frac { 1 } { 2 } \pi\).
   1. Find the set of values of \(\theta\) for which the progression is convergent.
   2. Find the exact value of the sum to infinity when \(\theta = \frac { 1 } { 6 } \pi\).

9

1. The first term of an arithmetic progression is - 2222 and the common difference is 17 . Find the value of the first positive term.
2. The first term of a geometric progression is \(\sqrt { } 3\) and the second term is \(2 \cos \theta\), where \(0 < \theta < \pi\). Find the set of values of \(\theta\) for which the progression is convergent.

5

1. In a geometric progression, the sum to infinity is equal to eight times the first term. Find the common ratio.
2. In an arithmetic progression, the fifth term is 197 and the sum of the first ten terms is 2040. Find the common difference.

1. A new mineral has been discovered and is going to be mined over a number of years.

A model predicts that the mass of the mineral mined each year will decrease by \(15 \%\) per year, so that the mass of the mineral mined each year forms a geometric sequence. Given that the mass of the mineral mined during year 1 is 8000 tonnes,

1. show that, according to the model, the mass of the mineral mined during year 6 will be approximately 3550 tonnes. According to the model, there is a limit to the total mass of the mineral that can be mined.
2. With reference to the geometric series, state why this limit exists.
3. Calculate the value of this limit. It is decided that a total mass of 40000 tonnes of the mineral is required. This is going to be mined from year 1 to year \(N\) inclusive.
4. Assuming the model, find the value of \(N\).
   

10 For each of the following sequences, state with a reason whether it is convergent, periodic or neither. Each sequence continues in the pattern established by the given terms.

1. \(3 , \frac { 3 } { 2 } , \frac { 3 } { 4 } , \frac { 3 } { 8 } , \ldots\)
2. \(3,7,11,15 , \ldots\)
3. \(3,5 , - 3 , - 5,3,5 , - 3 , - 5 , \ldots\)

4 Sequences A, B and C are shown below. They each continue in the pattern established by the given terms.

1. Which of these sequences is periodic?
2. Which of these sequences is convergent?
3. Find, in terms of \(n\), the \(n\)th term of sequence A .

9

1. An arithmetic progression has first term \(\log _ { 2 } 27\) and common difference \(\log _ { 2 } x\).
   1. Show that the fourth term can be written as \(\log _ { 2 } \left( 27 x ^ { 3 } \right)\).
   2. Given that the fourth term is 6, find the exact value of \(x\).
2. A geometric progression has first term \(\log _ { 2 } 27\) and common ratio \(\log _ { 2 } y\).
   1. Find the set of values of \(y\) for which the geometric progression has a sum to infinity.
   2. Find the exact value of \(y\) for which the sum to infinity of the geometric progression is 3 . \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}

15. An infinite geometric series has first four terms \(1 - 2 x + 4 x ^ { 2 } - 8 x ^ { 3 } + \cdots\). The series is convergent.
a. Find the set of possible values of \(x\) for which the series converges. Given that \(\sum _ { r = 1 } ^ { \infty } ( - 2 x ) ^ { r - 1 } = 8\),
b. calculate the value of \(x\).

The common ratio of a geometric progression is \(r\). The first term of the progression is \((r^2 - 3r + 2)\) and the sum to infinity is \(S\).

1. Show that \(S = 2 - r\). [2]
2. Find the set of possible values that \(S\) can take. [2]

The first term of a geometric series is 8. The sum to infinity of the series is 10. Find the common ratio. [3]

A geometric progression has 6 as its first term. Its sum to infinity is 5. Calculate its common ratio. [3]

Helen is creating a mosaic pattern by placing square tiles next to each other along a straight line. \includegraphics{figure_9} The area of each tile is half the area of the previous tile, and the sides of the largest tile have length \(w\) centimetres.

1. Find, in terms of \(w\), the length of the sides of the second largest tile. [1 mark]
2. Assume the tiles are in contact with adjacent tiles, but do not overlap. Show that, no matter how many tiles are in the pattern, the total length of the series of tiles will be less than \(3.5w\). [4 marks]
3. Helen decides the pattern will look better if she leaves a 3 millimetre gap between adjacent tiles. Explain how you could refine the model used in part (b) to account for the 3 millimetre gap, and state how the total length of the series of tiles will be affected. [2 marks]

Each of the series below shows the first four terms of a geometric series. Identify the only one of these geometric series that is convergent. [1 mark] Tick (\(\checkmark\)) one box. \(0.1 + 0.2 + 0.4 + 0.8 + \ldots\) \(1 - 1 + 1 - 1 + \ldots\) \(128 - 64 + 32 - 16 + \ldots\) \(1 + 2 + 4 + 8 + \ldots\)

In this question you must show detailed reasoning. It is given that the geometric series $$1 + \frac{5}{3x-4} + \left(\frac{5}{3x-4}\right)^2 + \left(\frac{5}{3x-4}\right)^3 + \ldots$$ is convergent.

1. Find the set of possible values of \(x\), giving your answer in set notation. [5]
2. Given that the sum to infinity of the series is \(\frac{2}{3}\), find the value of \(x\). [3]