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

11 The first three terms of a geometric sequence are \(5 k - 2,3 k - 6 , k + 2\), where \(k\) is a constant.

1. Show that \(k\) satisfies the equation \(k ^ { 2 } - 11 k + 10 = 0\).
2. When \(k\) takes the smaller of the two possible values, find the sum of the first 20 terms of the sequence.
3. When \(k\) takes the larger of the two possible values, find the sum to infinity of the sequence.

4

1. The first four terms of a sequence are \(2,3,0,3\) and the subsequent terms are given by \(\mathrm { a } _ { \mathrm { k } + 4 } = \mathrm { a } _ { \mathrm { k } }\).
   1. State what type of sequence this is.
   2. Find \(\sum _ { \mathrm { k } = 1 } ^ { 200 } \mathrm { a } _ { \mathrm { k } }\).
2. A different sequence is given by \(\mathrm { u } _ { \mathrm { n } } = \mathrm { b } ^ { \mathrm { n } }\) where \(b\) is a constant and \(n \geqslant 1\).
   1. State the set of values of \(b\) for which this is a divergent sequence.
   2. In the case where \(b = \frac { 1 } { 3 }\), find the sum of all the terms in the sequence.

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

13 The population of Melchester is 185207. During a nationwide flu epidemic the number of new cases in Melchester are recorded each day. The results from the first three days are shown in Fig. 13. \begin{table}[h] \end{table} A doctor notices that the numbers of new cases on successive days are in geometric progression.

1. Find the common ratio for this geometric progression. The doctor uses this geometric progression to model the number of new cases of flu in Melchester.
2. According to the model, how many new cases will there be on day 5?
3. Find a formula for the total number of cases from day 1 to day \(n\) inclusive according to this model, simplifying your answer.
4. Determine the maximum number of days for which the model could be viable in Melchester.
5. State, with a reason, whether it is likely that the model will be viable for the number of days found in part (d).

1 Determine the sum of the infinite geometric series \(9 - 3 + 1 - \frac { 1 } { 3 } + \frac { 1 } { 9 } + \ldots\)

15

1. Show that \(\sum _ { r = 1 } ^ { \infty } 0.99 ^ { r - 1 } \times 0.01 = 1\). Kofi is a very good table tennis player. Layla is determined to beat him.
   Every week they play one match of table tennis against each other. They will stop playing when Layla wins the match for the first time. \(X\) is the discrete random variable "the number of matches they play in total". Kofi models the situation using the probability function \(\mathrm { P } ( \mathrm { X } = \mathrm { r } ) = 0.99 ^ { \mathrm { r } - 1 } \times 0.01 \quad r = 1,2,3,4 , \ldots\) Kofi states that he is \(95 \%\) certain that Layla will not beat him within 6 years.
2. Determine whether Kofi's statement is consistent with his model. In between matches, Layla practises, but Kofi does not.
3. Explain why Layla might disagree with Kofi's model. Layla models the situation using the probability function \(\mathrm { P } ( \mathrm { X } = \mathrm { r } ) = \mathrm { kr } ^ { 2 } \quad r = 1,2,3,4,5,6,7,8\).
4. Explain how Layla's model takes into account the fact that she practises between matches, but Kofi's does not. Layla states that she is \(95 \%\) certain that she will beat Kofi within the first 6 matches.
5. Determine whether Layla's statement is consistent with her model.

3 A particular phone battery will last 10 hours when it is first used. Every time it is recharged, it will only last \(98 \%\) of its previous time. Find the maximum total length of use for the battery.

3

1. Use logarithms to solve the equation \(0.8 ^ { x } = 0.05\), giving your answer to three decimal places.
2. An infinite geometric series has common ratio \(r\). The sum to infinity of the series is five times the first term of the series.
   1. Show that \(r = 0.8\).
   2. Given that the first term of the series is 20 , find the least value of \(n\) such that the \(n\)th term of the series is less than 1 .

6 A geometric series has third term 36 and sixth term 972.

1. 1. Show that the common ratio of the series is 3 .
   2. Find the first term of the series.
2. The \(n\)th term of the series is \(u _ { n }\).
   1. Show that \(\sum _ { n = 1 } ^ { 20 } u _ { n } = 2 \left( 3 ^ { 20 } - 1 \right)\).
   2. Find the least value of \(n\) such that \(u _ { n } > 4 \times 10 ^ { 15 }\). \(7 \quad\) A curve \(C\) is defined for \(x > 0\) by the equation \(y = x + 3 + \frac { 8 } { x ^ { 4 } }\) and is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{1c06ba04-575c-4eb8-b4aa-0a7510838cd2-08_602_799_447_632}

6

1. A geometric series begins \(420 + 294 + 205.8 + \ldots\).
   1. Show that the common ratio of the series is 0.7 .
   2. Find the sum to infinity of the series.
   3. Write the \(n\)th term of the series in the form \(p \times q ^ { n }\), where \(p\) and \(q\) are constants.
2. The first term of an arithmetic series is 240 and the common difference of the series is - 8 . The \(n\)th term of the series is \(u _ { n }\).
   1. Write down an expression for \(u _ { n }\).
   2. Given that \(u _ { k } = 0\), find the value of \(\sum _ { n = 1 } ^ { k } u _ { n }\).

5 The sum to infinity of a geometric series is four times the first term of the series.

1. Show that the common ratio, \(r\), of the geometric series is \(\frac { 3 } { 4 }\).
2. The first term of the geometric series is 48 . Find the sum of the first 10 terms of the series, giving your answer to four decimal places.
3. The \(n\)th term of the geometric series is \(u _ { n }\) and the ( \(2 n\) )th term of the series is \(u _ { 2 n }\).
   1. Write \(u _ { n }\) and \(u _ { 2 n }\) in terms of \(n\).
   2. Hence show that \(\log _ { 10 } \left( u _ { n } \right) - \log _ { 10 } \left( u _ { 2 n } \right) = n \log _ { 10 } \left( \frac { 4 } { 3 } \right)\).
   3. Hence show that the value of $$\log _ { 10 } \left( \frac { u _ { 100 } } { u _ { 200 } } \right)$$ is 12.5 correct to three significant figures.

3 A geometric series begins $$20 + 16 + 12.8 + 10.24 + \ldots$$

1. Find the common ratio of the series.
2. Find the sum to infinity of the series.
3. Find the sum of the first 20 terms of the series, giving your answer to three decimal places.
4. Prove that the \(n\)th term of the series is \(25 \times 0.8 ^ { n }\).

5

1. An infinite geometric series has common ratio \(r\).
   The first term of the series is 10 and its sum to infinity is 50 .
   1. Show that \(r = \frac { 4 } { 5 }\).
   2. Find the second term of the series.
2. The first and second terms of the geometric series in part (a) have the same values as the 4th and 8th terms respectively of an arithmetic series.
   1. Find the common difference of the arithmetic series.
   2. The \(n\)th term of the arithmetic series is \(u _ { n }\). Find the value of \(\sum _ { n = 1 } ^ { 40 } u _ { n }\).

9 The first term of a geometric series is 12 and the common ratio of the series is \(\frac { 3 } { 8 }\).

1. Find the sum to infinity of the series.
2. Show that the sixth term of the series can be written in the form \(\frac { 3 ^ { 6 } } { 2 ^ { 13 } }\).
3. The \(n\)th term of the series is \(u _ { n }\).
   1. Write down an expression for \(u _ { n }\) in terms of \(n\).
   2. Hence show that $$\log _ { a } u _ { n } = n \log _ { a } 3 - ( 3 n - 5 ) \log _ { a } 2$$ (4 marks)

4 The \(n\)th term of a geometric series is \(u _ { n }\), where \(u _ { n } = 48 \left( \frac { 1 } { 4 } \right) ^ { n }\).

1. Find the value of \(u _ { 1 }\) and the value of \(u _ { 2 }\).
2. Find the value of the common ratio of the series.
3. Find the sum to infinity of the series.
4. Find the value of \(\sum _ { n = 4 } ^ { \infty } u _ { n }\).

1 A geometric series has first term 80 and common ratio \(\frac { 1 } { 2 }\).

1. Find the third term of the series.
2. Find the sum to infinity of the series.
3. Find the sum of the first 12 terms of the series, giving your answer to two decimal places.

3 The first term of a geometric series is 54 and the common ratio of the series is \(\frac { 8 } { 9 }\).

1. Find the sum to infinity of the series.
2. Find the second term of the series.
3. Show that the 12th term of the series can be written in the form \(\frac { 2 ^ { p } } { 3 ^ { q } }\), where \(p\) and \(q\) are integers.
   [0pt] [3 marks]

3 The first term of an infinite geometric series is 48 . The common ratio of the series is 0.6 .

1. Find the third term of the series.
2. Find the sum to infinity of the series.
3. The \(n\)th term of the series is \(u _ { n }\). Find the value of \(\sum _ { n = 4 } ^ { \infty } u _ { n }\).

7. A student completes a mathematics course and begins to work through past exam papers. He completes the first paper in 2 hours and the second in 1 hour 54 minutes. Assuming that the times he takes to complete successive papers form a geometric sequence,

1. find, to the nearest minute, how long he will take to complete the fifth paper,
2. show that the total time he takes to complete the first eight papers is approximately 13 hours 28 minutes,
3. find the least number of papers he must work through if he is to complete a paper in less than one hour.

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.

9. \begin{figure}[h] \end{figure} Figure 3 shows part of a design being produced by a computer program.
The program draws a series of circles with each one touching the previous one and such that their centres lie on a horizontal straight line. The radii of the circles form a geometric sequence with first term 1 mm and second term 1.5 mm . The width of the design is \(w\) as shown.

1. Find the radius of the fourth circle to be drawn.
2. Show that when eight circles have been drawn, \(w = 98.5 \mathrm {~mm}\) to 3 significant figures.
3. Find the total area of the design in square centimetres when ten circles have been drawn.

5 Prove by induction that, for all positive integers \(n , \sum _ { r = 1 } ^ { n } \frac { 1 } { 3 ^ { r } } = \frac { 1 } { 2 } \left( 1 - \frac { 1 } { 3 ^ { n } } \right)\).

13 The complex number \(z\) is defined as \(z = \frac { 1 } { 3 } \mathrm { e } ^ { \mathrm { i } \theta }\) where \(0 < \theta < \frac { 1 } { 2 } \pi\).
On an Argand diagram, the point O represents the complex number 0 , and the points \(P _ { 1 } , P _ { 2 } , P _ { 3 } , \ldots\) represent the complex numbers \(z , z ^ { 2 } , z ^ { 3 } , \ldots\) respectively.

1. Write down each of the following.
   1. The ratio of the lengths \(\mathrm { OP } _ { n + 1 } : \mathrm { OP } _ { n }\)
   2. The angle \(\mathrm { P } _ { n + 1 } \mathrm { OP } _ { n }\)
2. 1. Show that \(\left( 3 - \mathrm { e } ^ { \mathrm { i } \theta } \right) \left( 3 - \mathrm { e } ^ { - \mathrm { i } \theta } \right) = \mathrm { a } + \mathrm { b } \cos \theta\), where \(a\) and \(b\) are integers to be determined.
   2. By considering the sum to infinity of the series \(z + z ^ { 2 } + z ^ { 3 } + \ldots\), show that $$\frac { 1 } { 3 } \sin \theta + \frac { 1 } { 9 } \sin 2 \theta + \frac { 1 } { 27 } \sin 3 \theta + \ldots = \frac { 3 \sin \theta } { 10 - 6 \cos \theta } .$$

7 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 \(\sum _ { n = 1 } ^ { \infty } u _ { n } - \sum _ { n = 1 } ^ { N } u _ { n } < 10 ^ { - 4 }\).

11 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 \(g _ { 1 } = a _ { 1 } = 1 + \sqrt { 5 } , g _ { 3 } = a _ { 2 }\) and \(g _ { 4 } + a _ { 3 } = 0\). Given also that the geometric progression is convergent, show that its sum to infinity is \(4 + 2 \sqrt { 5 }\).