Geometric Sequences and Series

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

The first term of a geometric series is 5.4 and the common ratio is 0.1.

1. Find the fourth term of the series. [1]
2. Find the sum to infinity of the series. [2]

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]

3 Each year a company gives a grant to a charity. The amount given each year increases by \(5 \%\) of its value in the preceding year. The grant in 2001 was \(\\) 5000$. Find

1. the grant given in 2011,
2. the total amount of money given to the charity during the years 2001 to 2011 inclusive.

3 Each year a company gives a grant to a charity. The amount given each year increases by \(5 \%\) of its value in the preceding year. The grant in 2001 was \(\\) 5000$. Find

1. the grant given in 2011,
2. the total amount of money given to the charity during the years 2001 to 2011 inclusive.

4 There is a flowerhead at the end of each stem of an oleander plant. The next year, each flowerhead is replaced by three stems and flowerheads, as shown in Fig. 11. \begin{figure}[h] \end{figure}

1. How many flowerheads are there in year 5 ?
2. How many flowerheads are there in year \(n\) ?
3. As shown in Fig. 11, the total number of stems in year 2 is 4, (that is, 1 old one and 3 new ones). Similarly, the total number of stems in year 3 is 13 , (that is, \(1 + 3 + 9\) ). Show that the total number of stems in year \(n\) is given by \(\frac { 3 ^ { n } - 1 } { 2 }\).
4. Kitty's oleander has a total of 364 stems. Find
   (A) its age,
   (B) how many flowerheads it has.
5. Abdul's oleander has over 900 flowerheads. Show that its age, \(y\) years, satisfies the inequality \(y > \frac { \log _ { 10 } 900 } { \log _ { 10 } 3 } + 1\).
   Find the smallest integer value of \(y\) for which this is true.

2. The first three terms of a geometric series are ( \(p - 1\) ), 2 and ( \(2 p + 5\) ) respectively, where \(p\) is a constant. Find the two possible values of \(p\).

8

1. Over a 21-day period an athlete prepares for a marathon by increasing the distance she runs each day by 1.2 km . On the first day she runs 13 km .
   1. Find the distance she runs on the last day of the 21-day period.
   2. Find the total distance she runs in the 21-day period.
2. The first, second and third terms of a geometric progression are \(x , x - 3\) and \(x - 5\) respectively.
   1. Find the value of \(x\).
   2. Find the fourth term of the progression.
   3. Find the sum to infinity of the progression.

6

1. Given that the numbers \(a , b\) and \(c\) are in arithmetic progression, show that \(a + c = 2 b\).
2. Find an analogous result for three numbers in geometric progression.
3. The numbers \(2 - 3 x , 2 x , 3 - 2 x\) are the first three terms of a convergent geometric progression. Find \(x\) and hence calculate the sum to infinity.

4 An artist is creating a design for a large painting. The design includes a set of steps of varying heights. In the painting the lowest step has height 20 cm and the height of each other step is \(5 \%\) less than the height of the step immediately below it. In the painting the total height of the steps is 205 cm , correct to the nearest centimetre. Determine the number of steps in the design.

A trading company made a profit of £50 000 in 2006 (Year 1). A model for future trading predicts that profits will increase year by year in a geometric sequence with common ratio \(r, r > 1\). The model therefore predicts that in 2007 (Year 2) a profit of £50 000r will be made.

1. Write down an expression for the predicted profit in Year \(n\). [1]

The model predicts that in Year \(n\), the profit made will exceed £200 000.

1. Show that \(n > \frac{\log 4}{\log r} + 1\). [3]

Using the model with \(r = 1.09\),

1. find the year in which the profit made will first exceed £200 000, [2]
2. find the total of the profits that will be made by the company over the 10 years from 2006 to 2015 inclusive, giving your answer to the nearest £10 000. [3]

9. The resident population of a city is 130000 at the end of Year 1 A model predicts that the resident population of the city will increase by \(2 \%\) each year, with the populations at the end of each year forming a geometric sequence.

1. Show that the predicted resident population at the end of Year 2 is 132600
2. Write down the value of the common ratio of the geometric sequence. The model predicts that Year \(N\) will be the first year which will end with the resident population of the city exceeding 260000
3. Show that $$N > \frac { \log _ { 10 } 2 } { \log _ { 10 } 1.02 } + 1$$
4. Find the value of \(N\).
   

3 The common ratio of a geometric progression is 0.99 . Express the sum of the first 100 terms as a percentage of the sum to infinity, giving your answer correct to 2 significant figures.

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]

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]

An arithmetic progression (AP) and a geometric progression (GP) have the same first and fourth terms as each other. The first term of both is 1.5 and the fourth term of both is 12. Calculate the difference between the tenth terms of the AP and the GP. [5]

7 The second term of a geometric progression is 3 and the sum to infinity is 12 .

1. Find the first term of the progression. An arithmetic progression has the same first and second terms as the geometric progression.
2. Find the sum of the first 20 terms of the arithmetic progression.

10 An arithmetic sequence and a geometric sequence have \(n\)th terms \(a _ { n }\) and \(g _ { n }\) respectively, where \(n = 1,2,3 , \ldots\). It is given that \(a _ { 1 } = g _ { 1 } , a _ { 2 } = g _ { 2 } , a _ { 5 } = g _ { 3 } , a _ { 1 } \neq a _ { 2 }\) and \(a _ { 1 } \neq 0\).

1. Show that the common ratio of the geometric sequence is 3 .
2. Find the common difference of the arithmetic sequence in terms of \(a _ { 1 }\).
3. Let \(a _ { 1 } = g _ { 1 } = 5\).
   1. Find the first three terms of both sequences.
   2. Show that every term of the geometric sequence is also a term of the arithmetic sequence.

2 A geometric series has first term 3. The sum to infinity of the series is 8 .
Find the common ratio.

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

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]

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

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. \(0.1 + 0.2 + 0.4 + 0.8 + \ldots\) \(1 - 1 + 1 - 1 + \ldots\) \(128 - 64 + 32 - 16 + \ldots\) \(1 + 2 + 4 + 8 + \ldots\)

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 \(\theta = \frac { 1 } { 6 } \pi\).

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]

11. Wheat is to be grown on a farm. A model predicts that the mass of wheat harvested on the farm will increase by \(1.5 \%\) per year, so that the mass of wheat harvested each year forms a geometric sequence. Given that the mass of wheat harvested during year one is 6000 tonnes,

1. show that, according to the model, the mass of wheat harvested on the farm during year 4 will be approximately 6274 tonnes. During year \(N\), according to the model, there is predicted to be more than 8000 tonnes of wheat harvested on the farm.
2. Find the smallest possible value of \(N\). It costs \(\pounds 5\) per tonne to harvest the wheat.
3. Assuming the model, find the total amount that it would cost to harvest the wheat from year one to year 10 inclusive. Give your answer to the nearest \(\pounds 1000\).

1. There were 2100 tonnes of wheat harvested on a farm during 2017.

The mass of wheat harvested during each subsequent year is expected to increase by \(1.2 \%\) per year.

1. Find the total mass of wheat expected to be harvested from 2017 to 2030 inclusive, giving your answer to 3 significant figures. Each year it costs
   • £5.15 per tonne to harvest the first 2000 tonnes of wheat
   • £6.45 per tonne to harvest wheat in excess of 2000 tonnes
   • Use this information to find the expected cost of harvesting the wheat from 2017 to 2030 inclusive. Give your answer to the nearest \(\pounds 1000\)

1. Prove that the sum of the first \(n\) terms of a geometric series with first term \(a\) and common ratio \(r\) is given by $$\frac{a(1-r^n)}{1-r}.$$ [4]
2. Evaluate \(\sum_{r=1}^{12} (5 \times 2^r)\). [5]

9. (a) A geometric series has first term \(a\) and common ratio \(r\). Prove that the sum of the first \(n\) terms of the series is $$\frac { a \left( 1 - r ^ { n } \right) } { 1 - r } .$$ Mr. King will be paid a salary of \(\pounds 35000\) in the year 2005 . Mr. King's contract promises a \(4 \%\) increase in salary every year, the first increase being given in 2006, so that his annual salaries form a geometric sequence.
(b) Find, to the nearest \(\pounds 100\), Mr. King's salary in the year 2008. Mr. King will receive a salary each year from 2005 until he retires at the end of 2024.
(c) Find, to the nearest \(\pounds 1000\), the total amount of salary he will receive in the period from 2005 until he retires at the end of 2024.

A geometric series is \(a + ar + ar^2 + \ldots\)

1. Prove that the sum of the first \(n\) terms of this series is \(S_n = \frac{a(1 - r^n)}{1 - r}\). [4]

The first and second terms of a geometric series \(G\) are 10 and 9 respectively.

1. Find, to 3 significant figures, the sum of the first twenty terms of \(G\). [3]
2. Find the sum to infinity of \(G\). [2]

Another geometric series has its first term equal to its common ratio. The sum to infinity of this series is 10.

1. Find the exact value of the common ratio of this series. [3]

1. Evaluate

$$\sum _ { r = 1 } ^ { 12 } \left( 5 \times 2 ^ { r } \right)$$

2 Find

1. the sum of the first ten terms of the geometric progression \(81,54,36 , \ldots\),
2. the sum of all the terms in the arithmetic progression \(180,175,170 , \ldots , 25\).

The common ratio of a geometric progression is 0.99. Express the sum of the first 100 terms as a percentage of the sum to infinity, giving your answer correct to 2 significant figures. [5]

9. The sum to infinity of the geometric series $$a + a r + a r ^ { 2 } + \ldots$$ is 10 .
The sum to infinity of the series formed by the squares of the terms is 100/9.
a) Show that \(r = 4 / 5\) and find \(a\).
b) Find the sum to infinity of the series formed by the cubes of the terms. \section*{(Total for Question 9 is 5 marks)}

1. A geometric series has first term \(a\) and common ratio \(r\). The second term of the series is 4 and the sum to infinity of the series is 25.
   1. Show that \(25 r ^ { 2 } - 25 r + 4 = 0\).
   2. Find the two possible values of \(r\).
   3. Find the corresponding two possible values of \(a\).
   4. Show that the sum, \(S _ { n }\), of the first \(n\) terms of the series is given by
   $$S _ { n } = 25 \left( 1 - r ^ { n } \right) .$$ Given that \(r\) takes the larger of its two possible values,
2. find the smallest value of \(n\) for which \(S _ { n }\) exceeds 24 .

11. A geometric sequence, \(S _ { 1 }\), has first term \(a\) and common ratio \(r\) where \(a \neq 0\) and \(r \in ( - 1,1 )\) A new sequence, \(S _ { 2 }\), is formed by squaring each term of \(S _ { 1 }\)

1. Given that the sum to infinity of \(S _ { 2 }\) is twice the sum to infinity of \(S _ { 1 }\), show that \(a = 2 ( 1 + r )\) Fully justify your answer.
2. Determine the set of possible values for \(a\). \section*{Additional Answer Space } \section*{Additional Answer Space }

5. A geometric series has third term 36 and fourth term 27. Find

1. the common ratio of the series,
2. the fifth term of the series,
3. the sum to infinity of the series.

3 A progression has a second term of 96 and a fourth term of 54. Find the first term of the progression in each of the following cases:

1. the progression is arithmetic,
2. the progression is geometric with a positive common ratio.

9. The second and fifth terms of a geometric series are - 48 and 6 respectively.

1. Find the first term and the common ratio of the series.
2. Find the sum to infinity of the series.
3. Show that the difference between the sum of the first \(n\) terms of the series and its sum to infinity is given by \(2 ^ { 6 - n }\).

4 A geometric progression is such that the third term is 1764 and the sum of the second and third terms is 3444 . Find the 50th term.

3 A geometric sequence has a sum to infinity of - 3 A second sequence is formed by multiplying each term of the original sequence by - 2 What is the sum to infinity of the new sequence? Circle your answer. The sum to infinity does not

1 A geometric progression \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(u _ { 1 } = 32\) and \(u _ { n + 1 } = 0.75 u _ { n }\) for \(n \geqslant 1\).

1. Find \(u _ { 5 }\).
2. Find \(\sum _ { n = 1 } ^ { \infty } u _ { n }\).

4 The circumference round the trunk of a large tree is measured and found to be 5.00 m . After one year the circumference is measured again and found to be 5.02 m .

1. Given that the circumferences at yearly intervals form an arithmetic progression, find the circumference 20 years after the first measurement.
2. Given instead that the circumferences at yearly intervals form a geometric progression, find the circumference 20 years after the first measurement.

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

13 Consider a geometric sequence in which all the terms are positive real numbers. Show that, for any three consecutive terms of this sequence, the middle one is the geometric mean of the other two.

10. In a geometric series the common ratio is \(r\) and sum to \(n\) terms is \(S _ { n }\) Given $$S _ { \infty } = \frac { 8 } { 7 } \times S _ { 6 }$$ show that \(r = \pm \frac { 1 } { \sqrt { k } }\), where \(k\) is an integer to be found.