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

1. (i) (a) Find, in ascending powers of \(x\), the 2nd, 3rd and 5th terms of the binomial expansion of

$$( 3 + 2 x ) ^ { 6 }$$ For a particular value of \(x\), these three terms form consecutive terms in a geometric series.
(b) Find this value of \(x\).
(ii) In a different geometric series,

• the first term is \(\sin ^ { 2 } \theta\)
• the common ratio is \(2 \cos \theta\)
• the sum to infinity is \(\frac { 8 } { 5 }\) (a) Show that

$$5 \cos ^ { 2 } \theta - 16 \cos \theta + 3 = 0$$ (b) Hence find the exact value of the 2nd term in the series.

2. 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 \(S _ { \infty } - S _ { N } < 0.5\) 2. 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. The second and fourth terms of a geometric series are 7.2 and 5.832 respectively.

The common ratio of the series is positive.
For this series, find

1. the common ratio,
2. the first term,
3. the sum of the first 50 terms, giving your answer to 3 decimal places,
4. the difference between the sum to infinity and the sum of the first 50 terms, giving your answer to 3 decimal places.

6. A car was purchased for \(\pounds 18000\) on 1 st January. On 1st January each following year, the value of the car is \(80 \%\) of its value on 1st January in the previous year.

1. Show that the value of the car exactly 3 years after it was purchased is \(\pounds 9216\). The value of the car falls below \(\pounds 1000\) for the first time \(n\) years after it was purchased.
2. Find the value of \(n\). An insurance company has a scheme to cover the maintenance of the car. The cost is \(\pounds 200\) for the first year, and for every following year the cost increases by \(12 \%\) so that for the 3rd year the cost of the scheme is \(\pounds 250.88\)
3. Find the cost of the scheme for the 5th year, giving your answer to the nearest penny.
4. Find the total cost of the insurance scheme for the first 15 years.
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3. The second and fifth terms of a geometric series are 750 and - 6 respectively. Find

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

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.

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 \(k\).

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.

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 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 \(N\).

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 \(\sum _ { n = 1 } ^ { 20 } u _ { n }\).