1.04j Sum to infinity: convergent geometric series |r|<1

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]

The sum to infinity of a geometric series is 96 The first term of the series is less than 30 The second term of the series is 18

1. Find the first term and common ratio of the series. [5 marks]
2. 1. Show that the \(n\)th term of the series, \(u_n\), can be written as $$u_n = \frac{3^n}{2^{2n-5}}$$ [4 marks]
   2. Hence show that $$\log_3 u_n = n(1 - 2\log_3 2) + 5\log_3 2$$ [3 marks]

A building has a leaking roof and, while it is raining, water drips into a 12 litre bucket. When the rain stops, the bucket is one third full. Water continues to drip into the bucket from a puddle on the roof. In the first minute after the rain stops, 30 millilitres of water drips into the bucket. In each subsequent minute, the amount of water that drips into the bucket reduces by 2%. During the \(n\)th minute after the rain stops, the volume of water that drips into the bucket is \(W_n\) millilitres.

1. Find \(W_2\) [1 mark]
2. Explain why $$W_n = A \times 0.98^{n-1}$$ and state the value of \(A\). [2 marks]
3. Find the increase in the water in the bucket 15 minutes after the rain stops. Give your answer to the nearest millilitre. [2 marks]
4. Assuming it does not start to rain again, find the maximum amount of water in the bucket. [3 marks]
5. After several hours the water has stopped dripping. Give two reasons why the amount of water in the bucket is not as much as the answer found in part (d). [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. \(0.1 + 0.2 + 0.4 + 0.8 + \ldots\) \(1 - 1 + 1 - 1 + \ldots\) \(128 - 64 + 32 - 16 + \ldots\) \(1 + 2 + 4 + 8 + \ldots\)

Find the sum of the infinite series \(50 + 25 + 12.5 + 6.25 + \ldots\). [2]

A tree is 80 cm in height when it is planted. In the first year, the tree grows in height by 32 cm. In each subsequent year, the tree grows in height by 90% of the growth of the previous year.

1. Find the height of the tree 10 years after it was planted. [4]
2. Determine the maximum height of the tree. [2]

Showing all your working, evaluate

1. \(\sum_{r=3}^{50} (4r + 5)\) [4]
2. \(\sum_{r=2}^{\infty} \left(540 \times \left(\frac{1}{3}\right)^r\right)\). [3]

Aled decides to invest £1000 in a savings scheme on the first day of each year. The scheme pays 8% compound interest per annum, and interest is added on the last day of each year. The amount of savings, in pounds, at the end of the third year is given by $$1000 \times 1 \cdot 08 + 1000 \times 1 \cdot 08^2 + 1000 \times 1 \cdot 08^3$$ Calculate, to the nearest pound, the amount of savings at the end of thirty years. [5]

A company which makes batteries for electric cars has a 10-year plan for growth. • In year 1 the company will make 2600 batteries • In year 10 the company aims to make 12000 batteries In order to calculate the number of batteries it will need to make each year, from year 2 to year 9, the company considers the following model: *the number of batteries made will increase by the same percentage each year* Showing detailed reasoning, calculate the total number of batteries made from year 1 to year 10. [3]

In the question you must show detailed reasoning Given that \(\log_a x = \frac{\log_n x}{\log_n a}\), show that the sum of the infinite series, where \(n = 0,1,2...\), $$\log_2 e - \log_4 e + \log_{16} e - \cdots + (-1)^n \log_{2^{2^n}} e + \cdots$$ is $$\frac{1}{\ln(2\sqrt{2})}$$ [5] [Total marks: 65]

In this question you must show detailed reasoning. A sequence \(t_1, t_2, t_3 \ldots\) is defined by \(t_n = 5 - 2n\). Use an algebraic method to find the smallest value of \(N\) such that $$\sum_{n=1}^{\infty} 2^{t_n} - \sum_{n=1}^{N} 2^{t_n} < 10^{-8}$$ [8]

In this question you must show detailed reasoning. A sequence \(t_1, t_2, t_3 \ldots\) is defined by \(t_n = 25 \times 0.6^n\). Use an algebraic method to find the smallest value of \(N\) such that $$\sum_{n=1}^{\infty} t_n - \sum_{n=1}^{N} t_n < 10^{-4}$$ [8]

In this question you must show detailed reasoning. The \(n\)th term of a geometric progression is denoted by \(g_n\) and the \(n\)th term of an arithmetic progression is denoted by \(a_n\). It is given that \(g_1 = a_1 = 1 + \sqrt{5}\), \(g_2 = a_2\) and \(g_3 + a_3 = 0\). Given also that the geometric progression is convergent, show that its sum to infinity is \(4 + 2\sqrt{5}\). [12]

The first term of a geometric series is 120. The sum to infinity of the series is 480.

1. Show that the common ratio, \(r\), is \(\frac{3}{4}\). [3]
2. Find, to 2 decimal places, the difference between the 5th and 6th term. [2]
3. Calculate the sum of the first 7 terms. [2]

The sum of the first \(n\) terms of the series is greater than 300.

1. Calculate the smallest possible value of \(n\). [4]

A small company which makes batteries for electric cars has a 10 year plan for growth. In year 1 the company will make 2600 batteries. In year 10 the company aims to make 12000 batteries. In order to calculate the number of batteries it will need to make each year from year 2 to year 9, the company considers two models. Model A assumes that the number of batteries it will make each year will increase by the same number each year.

1. According to model A, determine the number of batteries the company will make in year 2. Give your answer to the nearest whole number of batteries. [3]

Model B assumes that the numbers of batteries it will make each year will increase by the same percentage each year.

1. According to model B, determine the number of batteries the company will make in year 2. Give your answer to the nearest 10 batteries. [3]

Sam calculates the total number of batteries made from year 1 to year 10 inclusive, using each of the two models.

1. Calculate the difference between the two totals, giving your answer to the nearest 100 batteries. [3]

A geometric series has second term 16 and fourth term 8 All the terms of the series are positive. The \(n\)th term of the series is \(u_n\) Find the exact value of \(\sum_{n=5}^{\infty} u_n\) [5 marks]

Show that $$\sum_{n=2}^{\infty} \left(\frac{1}{4}\right)^n \cos(180n)^{\circ} = \frac{9}{28}$$ [3]

A ball is released from rest from a height of 5 m and bounces repeatedly on horizontal ground. After hitting the ground for the first time, the ball rises to a maximum height of 3 m. In a model for the motion of the ball • the maximum height after each bounce is 60% of the previous maximum height • the motion takes place in a vertical line

1. Using the model
   1. show that the maximum height after the 3rd bounce is 1.08 m,
   2. find the total distance the ball travels from release to when the ball hits the ground for the 5th time.
   [3] According to the model, after the ball is released, there is a limit, \(D\) metres, to the total distance the ball will travel.
2. Find the value of \(D\) [2] With reference to the model,
3. give a reason why, in reality, the ball will not travel \(D\) metres in total. [1]

In this question you must show detailed reasoning. A sequence \(u_1, u_2, u_3 \ldots\) is defined by \(u_n = 25 \times 0.6^n\). Use an algebraic method to find the smallest value of \(N\) such that \(\sum_{n=1}^{\infty} u_n - \sum_{n=1}^{N} u_n < 10^{-4}\). [7]

The first term of a geometric progression is \(10\) and the common ratio is \(0.8\).

1. Find the fourth term. [2]
2. Find the sum of the first \(20\) terms, giving your answer correct to \(3\) significant figures. [2]
3. The sum of the first \(N\) terms is denoted by \(S_N\), and the sum to infinity is denoted by \(S_\infty\). Show that the inequality \(S_\infty - S_N < 0.01\) can be written as $$0.8^N < 0.0002,$$ and use logarithms to find the smallest possible value of \(N\). [6]

The first three terms of a geometric sequence are $$u_1 = 3k + 4 \quad u_2 = 12 - 3k \quad u_3 = k + 16$$ where \(k\) is a constant. Given that the sequence converges,

1. Find the value of \(k\), giving a reason for your answer. [4]
2. Find the value of \(\sum_{r=2}^{\infty} u_r\). [3]

The first three terms of a geometric sequence are $$u_1 = 3k + 4 \quad u_2 = 12 - 3k \quad u_3 = k + 16$$ where \(k\) is a constant. Given that the sequence converges,

1. Find the value of k, giving a reason for your answer. [4]
2. Find the value of \(\sum_{r=2}^{\infty} u_r\) [3]

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]