1.04i Geometric sequences: nth term and finite series sum

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 \(25 \times 0.8 ^ { n }\).

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 geometric series is \(a + a r + a r ^ { 2 } + \ldots\)

1. Prove that the sum of the first \(n\) terms of this series is given by \(S _ { n } = \frac { a \left( 1 - r ^ { n } \right) } { 1 - r }\). The second and fourth terms of the series are 3 and 1.08 respectively.
   Given that all terms in the series are positive, find
2. the value of \(r\) and the value of \(a\),
3. the sum to infinity of the series. \includegraphics[max width=\textwidth, alt={}, center]{1033051d-18bf-4734-a556-4c8e1c789992-4_764_1159_294_299} Fig. 2 shows part of the curve with equation \(y = x ^ { 3 } - 6 x ^ { 2 } + 9 x\). The curve touches the \(x\)-axis at \(A\) and has a maximum turning point at \(B\).
   1. Show that the equation of the curve may be written as \(y = x ( x - 3 ) ^ { 2 }\), and hence write down the coordinates of \(A\).
   2. Find the coordinates of \(B\). The shaded region \(R\) is bounded by the curve and the \(x\)-axis.
   3. Find the area of \(R\).

8. (a) An arithmetic series has first term \(a\) and common difference \(d\). Prove that the sum of the first \(n\) terms of the series is \(\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ]\). A company made a profit of \(\pounds 54000\) in the year 2001. A model for future performance assumes that yearly profits will increase in an arithmetic sequence with common difference \(\pounds d\). This model predicts total profits of \(\pounds 619200\) for the 9 years 2001 to 2009 inclusive.
(b) Find the value of \(d\). Using your value of \(d\),
(c) find the predicted profit for the year 2011. An alternative model assumes that the company's yearly profits will increase in a geometric sequence with common ratio 1.06 . Using this alternative model and again taking the profit in 2001 to be \(\pounds 54000\),
(d) find the predicted profit for the year 2011.

8. A geometric series has first term \(a\) and common ratio \(r\) where \(r > 1\). The sum of the first \(n\) terms of the series is denoted by \(S _ { n }\). Given that \(S _ { 4 } = 10 \times S _ { 2 }\),

1. find the value of \(r\). Given also that \(S _ { 3 } = 26\),
2. find the value of \(a\),
3. show that \(S _ { 6 } = 728\).

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.

9. The first three terms of a geometric series are \(( x - 2 ) , ( x + 6 )\) and \(x ^ { 2 }\) respectively.

1. Show that \(x\) must be a solution of the equation $$x ^ { 3 } - 3 x ^ { 2 } - 12 x - 36 = 0$$
2. Verify that \(x = 6\) is a solution of equation (I) and show that there are no other real solutions. Using \(x = 6\),
3. find the common ratio of the series,
4. find the sum of the first eight terms of the series.

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.

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. The second and third terms of a geometric series are \(\log _ { 3 } 4\) and \(\log _ { 3 } 16\) respectively.

1. Find the common ratio of the series.
2. Show that the first term of the series is \(\log _ { 3 } 2\).
3. Find, to 3 significant figures, the sum of the first six terms 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 A ball is dropped from a height of 2.5 m above a horizontal floor. The ball bounces repeatedly on the floor.

1. Find the speed of the ball when it first hits the floor.
2. The coefficient of restitution between the ball and the floor is \(e\).
   1. Show that the time taken between the first contact of the ball with the floor and the second contact of the ball with the floor is \(\frac { 10 e } { 7 }\) seconds.
   2. Find, in terms of \(e\), the time taken between the second contact and the third contact of the ball with the floor.
3. Find, in terms of \(e\), the total vertical distance travelled by the ball from when it is dropped until its third contact with the floor.
4. State a modelling assumption for answering this question, other than the ball being a particle.

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 \(4 + 2 \sqrt { 5 }\).

6 In this question you must show detailed reasoning.
A sequence \(S\) has terms \(u _ { 1 } , u _ { 2 } , u _ { 3 } \ldots\) defined by \(u _ { 1 } = 500\) and \(u _ { n + 1 } = 0.8 u _ { n }\).

1. State whether \(S\) is an arithmetic sequence or a geometric sequence, giving a reason for your answer.
2. Find \(u _ { 20 }\).
3. Find \(\sum _ { n = 1 } ^ { 20 } u _ { n }\).
4. Given that \(\sum _ { n = k } ^ { \infty } u _ { n } = 1024\), find the value of \(k\).

9. An arithmetic series has first term \(a\) and common difference \(d\).

1. Prove that the sum of the first \(n\) terms of the series is $$\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ] .$$ Sean repays a loan over a period of \(n\) months. His monthly repayments form an arithmetic sequence. He repays \(\pounds 149\) in the first month, \(\pounds 147\) in the second month, \(\pounds 145\) in the third month, and so on. He makes his final repayment in the \(n\)th month, where \(n > 21\).
2. Find the amount Sean repays in the 21st month. Over the \(n\) months, he repays a total of \(\pounds 5000\).
3. Form an equation in \(n\), and show that your equation may be written as $$n ^ { 2 } - 150 n + 5000 = 0$$
4. Solve the equation in part (c).
5. State, with a reason, which of the solutions to the equation in part (c) is not a sensible solution to the repayment problem.

5 The second term of a geometric series is 48 and the fourth term is 3 .

1. Show that one possible value for the common ratio, \(r\), of the series is \(- \frac { 1 } { 4 }\) and state the other value.
2. In the case when \(r = - \frac { 1 } { 4 }\), find:
   1. the first term;
   2. the sum to infinity of the series.

2 The \(n\)th term of a geometric sequence is \(u _ { n }\), where $$u _ { n } = 3 \times 4 ^ { n }$$

1. Find the value of \(u _ { 1 }\) and show that \(u _ { 2 } = 48\).
2. Write down the common ratio of the geometric sequence.
3. 1. Show that the sum of the first 12 terms of the geometric sequence is \(4 ^ { k } - 4\), where \(k\) is an integer.
   2. Hence find the value of \(\sum _ { n = 2 } ^ { 12 } u _ { n }\).