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

7

1. In an arithmetic progression, the first term is 12 and the sum of the first 70 terms is 12915 . Find the common difference.
2. In a geometric progression, the second term is - 4 and the sum to infinity is 9 . Find the common ratio.

5 The first three terms of a geometric progression are 4, 2, 1.
Find the twentieth term, expressing your answer as a power of 2.
Find also the sum to infinity of this progression.

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

8 The second term of a geometric progression is 18 and the fourth term is 2 . The common ratio is positive. Find the sum to infinity of this progression.

11 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.

11

1. André is playing a game where he makes piles of counters. He puts 3 counters in the first pile. Each successive pile he makes has 2 more counters in it than the previous one.
   1. How many counters are there in his sixth pile?
   2. André makes ten piles of counters. How many counters has he used altogether?
2. In another game, played with an ordinary fair die and counters, Betty needs to throw a six to start. The probability \(\mathrm { P } _ { n }\) of Betty starting on her \(n\)th throw is given by $$P _ { n } = \frac { 1 } { 6 } \times \left( \frac { 5 } { 6 } \right) ^ { n - 1 }$$
   1. Calculate \(\mathrm { P } _ { 4 }\). Give your answer as a fraction.
   2. The values \(\mathrm { P } _ { 1 } , \mathrm { P } _ { 2 } , \mathrm { P } _ { 3 } , \ldots\) form an infinite geometric progression. State the first term and the common ratio of this progression. Hence show that \(\mathrm { P } _ { 1 } + \mathrm { P } _ { 2 } + \mathrm { P } _ { 3 } + \ldots = 1\).
   3. Given that \(\mathrm { P } _ { n } < 0.001\), show that \(n\) satisfies the inequality $$n > \frac { \log _ { 10 } 0.006 } { \log _ { 10 } \left( \frac { 5 } { 6 } \right) } + 1$$ Hence find the least value of \(n\) for which \(\mathrm { P } _ { n } < 0.001\).

7 The first two terms of a geometric series are 5 and 4.
Find

1. the sum of the first 10 terms,
2. the sum to infinity.

3 The sum to infinity of a geometric series is 5 and the first term is 2 .
Find the common ratio of the series.

1 The common ratio of a geometric progression is - 0.5 . The sum of its first three terms is 15 . Find the first term.
Find also the sum to infinity.

11 When Fred joined a computer firm his salary was \(\pounds 28000\) per annum. In each subsequent year he received an annual increase of \(12 \%\) of his previous year's salary.

1. State Fred's salary for each of his first 3 years with the company. State also the common ratio of the geometric sequence formed by his salaries.
2. How much did Fred earn in the tenth year?
3. Show that the total amount Fred earned over the ten years was between \(\pounds 400000\) and £500000.
4. When Fred joined the computer firm, his brother Archie set up a plumbing business. He earned \(\pounds 35000\) in his first year and each year earned \(\pounds d\) more than in the previous year. At the end of ten years, he had earned exactly the same total amount as Fred. Calculate the value of \(d\).

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.
2. Find the sum to infinity of the series.

4

1. André is playing a game where he makes piles of counters. He puts 3 counters in the first pile. Each successive pile he makes has 2 more counters in it than the previous one.
   1. How many counters are there in his sixth pile?
   2. André makes ten piles of counters. How many counters has he used altogether?
2. In another game, played with an ordinary fair die and counters, Betty needs to throw a six to start. The probability \(\mathrm { P } _ { n }\) of Betty starting on her \(n\)th throw is given by $$P _ { n } = \frac { 1 } { 6 } \times \left( \frac { 5 } { 6 } \right) ^ { n - 1 }$$
   1. Calculate \(\mathrm { P } _ { 4 }\). Give your answer as a fraction.
   2. The values \(\mathrm { P } _ { 1 } , \mathrm { P } _ { 2 } , \mathrm { P } _ { 3 } , \ldots\) form an infinite geometric progression. State the first term and the common ratio of this progression. Hence show that \(\mathrm { P } _ { 1 } + \mathrm { P } _ { 2 } + \mathrm { P } _ { 3 } + \ldots = 1\).
   3. Given that \(\mathrm { P } _ { n } < 0.001\), show that \(n\) satisfies the inequality $$n > \frac { \log _ { 10 } 0.006 } { \log _ { 10 } \left( \frac { 5 } { 6 } \right) } + 1$$ Hence find the least value of \(n\) for which \(\mathrm { P } _ { n } < 0.001\).

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

2 The first three terms of a geometric progression are 4, 2, 1.
Find the twentieth term, expressing your answer as a power of 2 .
Find also the sum to infinity of this progression.

2 The terms of a sequence are given by $$\begin{aligned} u _ { 1 } & = 192 , \\ u _ { n + 1 } & = - \frac { 1 } { 2 } u _ { n } . \end{aligned}$$

1. Find the third term of this sequence and state what type of sequence it is.
2. Show that the series \(u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots\) converges and find its sum to infinity.

5

1. Show that $$\frac { 1 } { 2 r - 1 } - \frac { 1 } { 2 r + 1 } = \frac { 2 } { 4 r ^ { 2 } - 1 }$$
2. Hence find an expression in terms of \(n\) for $$\frac { 2 } { 3 } + \frac { 2 } { 15 } + \frac { 2 } { 35 } + \ldots + \frac { 2 } { 4 n ^ { 2 } - 1 }$$
3. State the value of
   1. \(\quad \sum _ { r = 1 } ^ { \infty } \frac { 2 } { 4 r ^ { 2 } - 1 }\),
   2. \(\quad \sum _ { r = n + 1 } ^ { \infty } \frac { 2 } { 4 r ^ { 2 } - 1 }\).

6.Given that the coefficients of \(x , x ^ { 2 }\) and \(x ^ { 4 }\) in the expansion of \(( 1 + k x ) ^ { n }\) ,where \(n \geq 4\) and \(k\) is a positive constant,are the consecutive terms of a geometric series,

1. show that \(k = \frac { 6 ( n - 1 ) } { ( n - 2 ) ( n - 3 ) }\) .
2. Given further that both \(n\) and \(k\) are positive integers,find all possible pairs of values for \(n\) and \(k\) .You should show clearly how you know that you have found all possible pairs of values.
3. For the case where \(k = 1.4\) ,find the value of the positive integer \(n\) .
4. Given that \(k = 1.4 , n\) is a positive integer and that the first term of the geometric series is the coefficient of \(x\) ,estimate how many terms are required for the sum of the geometric series to exceed \(1.12 \times 10 ^ { 12 }\) .[You may assume that \(\log _ { 10 } 4 \approx 0.6\) and \(\log _ { 10 } 5 \approx 0.7\) .]

6. \begin{figure}[h] \end{figure} Figure 2 shows the first few iterations in the construction of a curve, \(L\).
Starting with a straight line \(L _ { 0 }\) of length 4 , the middle half of this line is replaced by three sides of a trapezium above \(L _ { 0 }\) as shown, such that the length of each of these sides is \(\frac { 1 } { 4 }\) of the length of \(L _ { 0 }\) After the first iteration each line segment has length one.
In subsequent iterations, each line segment parallel to \(L _ { 0 }\) similarly has its middle half replaced by three sides of a trapezium above that line segment, with each side \(\frac { 1 } { 4 }\) the length of that line segment. Line segments in \(L _ { n }\) are either parallel to \(L _ { 0 }\) or are sloped.

1. Show that the length of \(L _ { 2 }\) is \(\frac { 23 } { 4 }\)
2. Write down the number of
   1. line segments in \(L _ { n }\) that are parallel to \(L _ { 0 }\)
   2. sloped line segments in \(L _ { 2 }\) that are not in \(L _ { 1 }\)
   3. new sloped line segments that are created by the ( \(n + 1\) )th iteration.
3. Hence find the length of \(L _ { n }\) as \(n \rightarrow \infty\) The area enclosed between \(L _ { 0 }\) and \(L _ { n }\) is \(A _ { n }\)
4. Find the value of \(A _ { 1 }\)
5. Find, in terms of \(n\), an expression for \(A _ { n + 1 } - A _ { n }\)
6. Hence find the value of \(A _ { n }\) as \(n \rightarrow \infty\) The same construction as described above is applied externally to the three sides of an equilateral triangle of side length \(a\).
   Given that the limit of the area of the resulting shape is \(26 \sqrt { 3 }\)
7. find the value of \(a\).

7. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve \(C\) with question $$y = \mathrm { e } ^ { - x } \sin x , \quad x \geq 0 .$$

1. Find the coordinates of the points \(P , Q\) and \(R\) where \(C\) cuts the positive axis.
2. Use integration by parts to show that $$\int \mathrm { e } ^ { - x } \sin x \mathrm {~d} x = - \frac { 1 } { 2 } \mathrm { e } ^ { - x } ( \sin x + \cos x ) + \text { constant }$$ The terms of the sequence \(A _ { 1 } , A _ { 2 } , \ldots , A _ { n } , \ldots\) represent areas between \(C\) and the \(x\)-axis for successive portions of \(C\) where \(y\) is positive.The area represented by \(A _ { 1 }\) and \(A _ { 2 }\) are shown in Figure 3.
3. Find an expression for \(A _ { n }\) in terms of \(n\) and \(\pi\) .
   (6)
4. Show that \(A _ { 1 } + A _ { 2 } + \ldots + A _ { n } + \ldots\) is a geometric series with sum to infinity $$\frac { \mathrm { e } ^ { \pi } } { 2 \left( \mathrm { e } ^ { \pi } - 1 \right) } .$$
5. Given that $$\int _ { 0 } ^ { \infty } \mathrm { e } ^ { - x } \sin x \mathrm {~d} x = \frac { 1 } { 2 }$$ find the exact value of $$\int _ { 0 } ^ { \infty } \left| e ^ { - x } \sin x \right| d x$$ and simplify your answer. END

7. \includegraphics[max width=\textwidth, alt={}, center]{0df09d8a-7478-4679-b117-128ee226db6a-5_648_1590_296_275} The circle \(C _ { 1 }\) has centre \(O\) and radius \(R\). The tangents \(A P\) and \(B P\) to \(C _ { 1 }\) meet at the point \(P\) and angle \(A P B = 2 \alpha , 0 < \alpha < \frac { \pi } { 2 }\). A sequence of circles \(C _ { 1 } , C _ { 2 } , \ldots , C _ { n } , \ldots\) is drawn so that each new circle \(C _ { n + 1 }\) touches each of \(C _ { n } , A P\) and \(B P\) for \(n = 1,2,3 , \ldots\) as shown in Figure 2. The centre of each circle lies on the line \(O P\).

1. Show that the radii of the circles form a geometric sequence with common ratio $$\frac { 1 - \sin \alpha } { 1 + \sin \alpha }$$
2. Find, in terms of \(R\) and \(\alpha\), the total area enclosed by all the circles, simplifying your answer. The area inside the quadrilateral \(P A O B\), not enclosed by part of \(C _ { 1 }\) or any of the other circles, is \(S\).
3. Show that $$S = R ^ { 2 } \left( \alpha + \cot \alpha - \frac { \pi } { 4 } \operatorname { cosec } \alpha - \frac { \pi } { 4 } \sin \alpha \right) .$$
4. Show that, as \(\alpha\) varies, $$\frac { \mathrm { d } S } { \mathrm {~d} \alpha } = R ^ { 2 } \cot ^ { 2 } \alpha \left( \frac { \pi } { 4 } \cos \alpha - 1 \right)$$
5. Find, in terms of \(R\), the least value of \(S\) for \(\frac { \pi } { 6 } \leq \alpha \leq \frac { \pi } { 4 }\).

5. \begin{figure}[h] \end{figure} Figure 1 shows part of a sequence \(S _ { 1 } , S _ { 2 } , S _ { 3 } , \ldots\), of model snowflakes. The first term \(S _ { 1 }\) consists of a single square of side \(a\). To obtain \(S _ { 2 }\), the middle third of each edge is replaced with a new square, of side \(\frac { a } { 3 }\), as shown in Figure 1 . Subsequent terms are obtained by replacing the middle third of each external edge of a new square formed in the previous snowflake, by a square \(\frac { 1 } { 3 }\) of the size, as illustrated by \(S _ { 3 }\) in Figure 1.

1. Deduce that to form \(S _ { 4 } , 36\) new squares of side \(\frac { a } { 27 }\) must be added to \(S _ { 3 }\).
2. Show that the perimeters of \(S _ { 2 }\) and \(S _ { 3 }\) are \(\frac { 20 a } { 3 }\) and \(\frac { 28 a } { 3 }\) respectively.
3. Find the perimeter of \(S _ { n }\).
4. Describe what happens to the perimeter of \(S _ { n }\) as \(n\) increases.
5. Find the areas of \(S _ { 1 } , S _ { 2 }\) and \(S _ { 3 }\).
6. Find the smallest value of the constant \(S\) such that the area of \(S _ { n } < S\), for all values of \(n\). \includegraphics[max width=\textwidth, alt={}, center]{f2290882-b9a4-43ec-a38f-c44d46477242-5_590_1041_283_588} Figure 2 shows a sketch of the curve \(C\) with equation \(y = \tan \frac { t } { 2 } , \quad 0 \leq t \leq \frac { \pi } { 2 }\).
   The point \(P\) on \(C\) has coordinates \(\left( x , \tan \frac { x } { 2 } \right)\).
   The vertices of rectangle \(R\) are at \(( x , 0 ) , \left( \frac { x } { 2 } , 0 \right) , \left( \frac { x } { 2 } , \tan \frac { x } { 2 } \right)\) and \(\left( x , \tan \frac { x } { 2 } \right)\) as shown in Figure 2.

3.The angle \(\theta , 0 < \theta < \frac { \pi } { 2 }\) ,satisfies $$\tan \theta \tan 2 \theta = \sum _ { r = 0 } ^ { \infty } 2 \cos ^ { r } 2 \theta$$

1. Show that \(\tan \theta = 3 ^ { p }\) ,where \(p\) is a rational number to be found.
2. Hence show that \(\frac { \pi } { 6 } < \theta < \frac { \pi } { 4 }\)

4.A sequence of positive integers \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) has \(r\) th term given by $$a _ { r } = 2 ^ { r } - 1$$

1. Write down the first 6 terms of this sequence.
2. Verify that \(a _ { r + 1 } = 2 a _ { r } + 1\)
3. Find \(\sum _ { r = 1 } ^ { n } a _ { r }\)
4. Show that \(\frac { 1 } { a _ { r + 1 } } < \frac { 1 } { 2 } \times \frac { 1 } { a _ { r } }\)
5. Hence show that \(1 + \frac { 1 } { 3 } + \frac { 1 } { 7 } + \frac { 1 } { 15 } + \frac { 1 } { 31 } + \ldots < 1 + \frac { 1 } { 3 } + \left( \frac { 1 } { 7 } + \frac { \frac { 1 } { 2 } } { 7 } + \frac { \frac { 1 } { 4 } } { 7 } + \ldots \right)\)
6. Show that \(\frac { 31 } { 21 } < \sum _ { r = 1 } ^ { \infty } \frac { 1 } { a _ { r } } < \frac { 34 } { 21 }\)

5.(a)Show that $$\sum _ { r = 0 } ^ { n } x ^ { - r } = \frac { x } { x - 1 } - \frac { x ^ { - n } } { x - 1 } \quad \text { where } x \neq 0 \text { and } x \neq 1$$ (b)Hence find an expression in terms of \(x\) and \(n\) for \(\sum _ { r = 0 } ^ { n } r x ^ { - ( r + 1 ) }\) for \(x \neq 0\) and \(x \neq 1\) Simplify your answer.
(c)Find \(\sum _ { r = 0 } ^ { n } \left( \frac { 3 + 5 r } { 2 ^ { r } } \right)\) Give your answer in the form \(a - \frac { b + c n } { 2 ^ { n } }\) ,where \(a , b\) and \(c\) are integers.

6 A geometric progression has first term 20 and common ratio 0.9.

1. Find the sum to infinity.
2. Find the sum of the first 30 terms.
3. Use logarithms to find the smallest value of \(p\) such that the \(p\) th term is less than 0.4 .