1.04i Geometric sequences: nth term and finite series sum

Liquid is kept in containers, which due to evaporation and ongoing chemical reactions, at the end of each month the volume of the liquid in these containers reduces by 10% compared with the volume at the start of the same month. One such container is filled up with 250 litres of liquid.

1. Show that the volume of the liquid in the container at the end of the second month is 202.5 litres. [1]
2. Find the volume of the liquid in the container at the end of the twelfth month. [2]

At the start of each month a new container is filled up with 250 litres of liquid, so that at the end of twelve months there are 12 containers with liquid.

1. Use an algebraic method to calculate the total amount of liquid in the 12 containers at the end of 12 months. [5]

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 seventh term of a geometric progression is equal to twice the fifth term. The sum of the first seven terms is 254 and the terms are all positive. Find the first term, showing that it can be written in the form \(p + q\sqrt{r}\) where \(p\), \(q\) and \(r\) are integers. [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]

The functions f and g are defined by $$\text{f}(x) = \frac{3}{2}\ln x \quad x > 0$$ $$\text{g}(x) = \frac{4x + 3}{2x + 1} \quad x > 0$$

1. Find gf(\(\text{e}^2\)) writing your answer in simplest form. [2]
2. Find the range of the function fg. [2]
3. Given that f(8) and f(2) are the second and third terms respectively of a geometric series, find the sum to infinity of this series, giving your answer in the form \(a \ln 2\) where \(a\) is rational. [4]

An arithmetic progression has first term \(a\) and common difference \(d\), where \(a\) and \(d\) are non-zero. The first, third and fourth terms of the arithmetic progression are consecutive terms of a geometric progression with common ratio \(r\).

1. 1. Show that \(r = \frac{a + 2d}{a}\). [1]
   2. Find \(d\) in terms of \(a\). [2]
2. Find the common ratio of the geometric progression. [2]

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. A geometric series has common ratio \(r\) and first term \(a\). Given \(r \neq 1\) and \(a \neq 0\)

1. prove that $$S_n = \frac{a(1-r^n)}{1-r}$$ [4]

Given also that \(S_{10}\) is four times \(S_5\)

1. find the exact value of \(r\). [4]

The first term of a geometric progression is 12 and the second term is 9.

1. Find the fifth term. [3]

The sum of the first \(N\) terms is denoted by \(S_N\) and the sum to infinity is denoted by \(S_\infty\). It is given that the difference between \(S_\infty\) and \(S_N\) is at most 0.0096.

1. Show that \(\left(\frac{3}{4}\right)^N \leqslant 0.0002\). [3]
2. Use logarithms to find the smallest possible value of \(N\). [2]

The table shows information about three geometric series. The three geometric series have different common ratios.

1. Find \(n_1\). [2]
2. Given that \(r_2\) is an integer less than 10, find the value of \(r_2\) and the value of \(n_2\). [2]
3. Given that \(r_3\) is not an integer, find a possible value for the sum of all the terms in Series 3. [4]

Business A made a £5000 profit during its first year. In each subsequent year, the profit increased by £1500 so that the profit was £6500 during the second year, £8000 during the third year and so on. Business B made a £5000 profit during its first year. In each subsequent year, the profit was 90% of the previous year's profit.

1. Find an expression for the total profit made by business A during the first \(n\) years. Give your answer in its simplest form. [2]
2. Find an expression for the total profit made by business B during the first \(n\) years. Give your answer in its simplest form. [3]
3. Find how many years it will take for the total profit of business A to reach £385 000. [3]
4. Comment on the profits made by each business in the long term. [2]

A geometric progression with common ratio \(r\) consists of positive terms. The sum of the first four terms is five times the sum of the first two terms.

1. Find an equation in \(r\) and deduce that \(r = 2\). [3]
2. Given that the fifth term is 192, find the value of the first term. [1]
3. Find the smallest value of \(n\) such that the sum of the first \(n\) terms of the progression exceeds \(10^{64}\). [3]

An arithmetic progression has first term \(a\) and common difference \(d\). The first, ninth and fourteenth terms are, respectively, the first three terms of a geometric progression with common ratio \(r\), where \(r \neq 1\).

1. Find \(d\) in terms of \(a\) and show that \(r = \frac{5}{3}\). [7]
2. Find the sum to infinity of the geometric progression in terms of \(a\). [2]

A sequence \(\{u_n\}\) is given by $$u_1 = k$$ $$u_{2n} = u_{2n-1} \times p \qquad n \geq 1$$ $$u_{2n+1} = u_{2n} \times q \qquad n \geq 1$$ where \(k\), \(p\) and \(q\) are positive constants with \(pq \neq 1\)

1. Write down the first 6 terms of this sequence. [3]
2. Show that \(\sum_{r=1}^{2n} u_r = \frac{k(1+p)(1-(pq)^n)}{1-pq}\) [6]

In part (c) \([x]\) means the integer part of \(x\), so for example \([2.73] = 2\), \([4] = 4\) and \([0] = 0\)

1. Find \(\sum_{r=1}^{\infty} 6 \times \left(\frac{4}{3}\right)^{\left[\frac{r}{2}\right]} \times \left(\frac{3}{5}\right)^{\left[\frac{r-1}{2}\right]}\) [4]

[Total 13 marks]