1.04i Geometric sequences: nth term and finite series sum

1. A geometric series has first term \(a = 360\) and common ratio \(r = \frac { 7 } { 8 }\)

Giving your answers to 3 significant figures where appropriate, find

1. the 20 th term of the series,
2. the sum of the first 20 terms of the series,
3. the sum to infinity of the series.

3. A company predicts a yearly profit of \(\pounds 120000\) in the year 2013 . The company predicts that the yearly profit will rise each year by \(5 \%\). The predicted yearly profit forms a geometric sequence with common ratio 1.05

1. Show that the predicted profit in the year 2016 is \(\pounds 138915\)
2. Find the first year in which the yearly predicted profit exceeds \(\pounds 200000\)
3. Find the total predicted profit for the years 2013 to 2023 inclusive, giving your answer to the nearest pound.

4. The first term of a geometric series is 5 and the common ratio is 1.2 For this series find, to 1 decimal place,

1. 1. the \(20 ^ { \text {th } }\) term,
   2. the sum of the first 20 terms. The sum of the first \(n\) terms of the series is greater than 3000
2. Calculate the smallest possible value of \(n\).

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.

8. A trading company made a profit of \(\pounds 50000\) 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 \(\pounds 50000 r\) will be made.

1. Write down an expression for the predicted profit in Year \(n\). The model predicts that in Year \(n\), the profit made will exceed \(\pounds 200000\).
2. Show that \(n > \frac { \log 4 } { \log r } + 1\). Using the model with \(r = 1.09\),
3. find the year in which the profit made will first exceed \(\pounds 200000\),
4. 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 \(\pounds 10000\).

6. A geometric series has first term 5 and common ratio \(\frac { 4 } { 5 }\). Calculate

1. the 20th term of the series, to 3 decimal places,
2. the sum to infinity of the series. Given that the sum to \(k\) terms of the series is greater than 24.95,
3. show that \(k > \frac { \log 0.002 } { \log 0.8 }\),
4. find the smallest possible value of \(k\).

9. The adult population of a town is 25000 at the end of Year 1. A model predicts that the adult population of the town will increase by \(3 \%\) each year, forming a geometric sequence.

1. Show that the predicted adult population at the end of Year 2 is 25750.
2. Write down the common ratio of the geometric sequence. The model predicts that Year \(N\) will be the first year in which the adult population of the town exceeds 40000.
3. Show that $$( N - 1 ) \log 1.03 > \log 1.6$$
4. Find the value of \(N\). At the end of each year, each member of the adult population of the town will give \(\pounds 1\) to a charity fund. Assuming the population model,
5. find the total amount that will be given to the charity fund for the 10 years from the end of Year 1 to the end of Year 10, giving your answer to the nearest \(\pounds 1000\).

1. The second and third terms of a geometric series are 192 and 144 respectively.

For this series, find

1. the common ratio,
2. the first term,
3. the sum to infinity,
4. the smallest value of \(n\) for which the sum of the first \(n\) terms of the series exceeds 1000.

1. The first three terms of a geometric series are

$$18,12 \text { and } p$$ respectively, where \(p\) is a constant. Find

1. the value of the common ratio of the series,
2. the value of \(p\),
3. the sum of the first 15 terms of the series, giving your answer to 3 decimal places.

2. A geometric series has first term \(a\), where \(a \neq 0\), and common ratio \(r\). The sum to infinity of this series is 6 times the first term of the series.

1. Show that \(r = \frac { 5 } { 6 }\) Given that the fourth term of this series is 62.5
2. find the value of \(a\),
3. find the difference between the sum to infinity and the sum of the first 30 terms, giving your answer to 3 significant figures.

6. The first term of a geometric series is 20 and the common ratio is \(\frac { 7 } { 8 }\) The sum to infinity of the series is \(S _ { \infty }\)

1. Find the value of \(S _ { \infty }\) The sum to \(N\) terms of the series is \(S _ { N }\)
2. Find, to 1 decimal place, the value of \(S _ { 12 }\)
3. Find the smallest value of \(N\), for which $$S _ { \infty } - S _ { N } < 0.5$$

1. 1. All the terms of a geometric series are positive. The sum of the first two terms is 34 and the sum to infinity is 162

Find

1. the common ratio,
2. the first term.
   (ii) A different geometric series has a first term of 42 and a common ratio of \(\frac { 6 } { 7 }\). Find the smallest value of \(n\) for which the sum of the first \(n\) terms of the series exceeds 290

1. A geometric series has first term \(a\) and common ratio \(r = \frac { 3 } { 4 }\)

The sum of the first 4 terms of this series is 175

1. Show that \(a = 64\)
2. Find the sum to infinity of the series.
3. Find the difference between the 9th and 10th terms of the series. Give your answer to 3 decimal places.

9. The first three terms of a geometric sequence are $$7 k - 5,5 k - 7,2 k + 10$$ where \(k\) is a constant.

1. Show that \(11 k ^ { 2 } - 130 k + 99 = 0\) Given that \(k\) is not an integer,
2. show that \(k = \frac { 9 } { 11 }\) For this value of \(k\),
3. 1. evaluate the fourth term of the sequence, giving your answer as an exact fraction,
   2. evaluate the sum of the first ten terms of the sequence.

6. At the beginning of the year 2000 a company bought a new machine for \(\pounds 15000\). Each year the value of the machine decreases by \(20 \%\) of its value at the start of the year.

1. Show that at the start of the year 2002, the value of the machine was \(\pounds 9600\). When the value of the machine falls below \(\pounds 500\), the company will replace it.
2. Find the year in which the machine will be replaced. To plan for a replacement machine, the company pays \(\pounds 1000\) at the start of each year into a savings account. The account pays interest at a fixed rate of \(5 \%\) per annum. The first payment was made when the machine was first bought and the last payment will be made at the start of the year in which the machine is replaced.
3. Using your answer to part (b), find how much the savings account will be worth immediately after the payment at the start of the year in which the machine is replaced.

9 A geometric progression has first term \(a\), where \(a \neq 0\), and common ratio \(r\), where \(r \neq 1\). The difference between the fourth term and the first term is equal to four times the difference between the third term and the second term.

1. Show that \(r ^ { 3 } - 4 r ^ { 2 } + 4 r - 1 = 0\).
2. Show that \(r - 1\) is a factor of \(r ^ { 3 } - 4 r ^ { 2 } + 4 r - 1\). Hence factorise \(r ^ { 3 } - 4 r ^ { 2 } + 4 r - 1\).
3. Hence find the two possible values for the ratio of the geometric progression. Give your answers in an exact form.
4. For the value of \(r\) for which the progression is convergent, prove that the sum to infinity is \(\frac { 1 } { 2 } a ( 1 + \sqrt { } 5 )\).

5 In a geometric progression, the first term is 5 and the second term is 4.8 .

1. Show that the sum to infinity is 125 .
2. The sum of the first \(n\) terms is greater than 124 . Show that $$0.96 ^ { n } < 0.008$$ and use logarithms to calculate the smallest possible value of \(n\).

8 The first term of a geometric progression is 10 and the common ratio is 0.8.

1. Find the fourth term.
2. Find the sum of the first 20 terms, giving your answer correct to 3 significant figures.
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 \(N\).

8 The amounts of oil pumped from an oil well in each of the years 2001 to 2004 formed a geometric progression with common ratio 0.9 . The amount pumped in 2001 was 100000 barrels.

1. Calculate the amount pumped in 2004. It is assumed that the amounts of oil pumped in future years will continue to follow the same geometric progression. Production from the well will stop at the end of the first year in which the amount pumped is less than 5000 barrels.
2. Calculate in which year the amount pumped will fall below 5000 barrels.
3. Calculate the total amount of oil pumped from the well from the year 2001 up to and including the final year of production.

6

1. John aims to pay a certain amount of money each month into a pension fund. He plans to pay \(\pounds 100\) in the first month, and then to increase the amount paid by \(\pounds 5\) each month, i.e. paying \(\pounds 105\) in the second month, \(\pounds 110\) in the third month, etc. If John continues making payments according to this plan for 240 months, calculate
   1. how much he will pay in the final month,
   2. how much he will pay altogether over the whole period.
   3. Rachel also plans to pay money monthly into a pension fund over a period of 240 months, starting with \(\pounds 100\) in the first month. Her monthly payments will form a geometric progression, and she will pay \(\pounds 1500\) in the final month. Calculate how much Rachel will pay altogether over the whole period.

1 A geometric progression \(\mathrm { u } _ { 1 } , \mathrm { u } _ { 2 } , \mathrm { u } _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 15 \quad \text { and } \quad u _ { n + 1 } = 0.8 u _ { n } \text { for } n \geqslant 1 .$$

1. Write down the values of \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\).
2. Find \(\sum _ { n = 1 } ^ { 20 } u _ { n }\).