1.04i Geometric sequences: nth term and finite series sum

4 Sequences A, B and C are shown below. They each continue in the pattern established by the given terms.

1. Which of these sequences is periodic?
2. Which of these sequences is convergent?
3. Find, in terms of \(n\), the \(n\)th term of sequence A .

1.(a)By considering the series $$1 + t + t ^ { 2 } + t ^ { 3 } + \ldots + t ^ { n }$$ or otherwise,sum the series $$1 + 2 t + 3 t ^ { 2 } + 4 t ^ { 3 } + \ldots + n t ^ { n - 1 }$$ for \(t \neq 1\) .
(b)Hence find and simplify an expression for $$1 + 2 \times 3 + 3 \times 3 ^ { 2 } + 4 \times 3 ^ { 3 } + \ldots + 2001 \times 3 ^ { 2000 }$$ (c)Write down an expression for both the sums of the series in part(a)for the case where \(t = 1\) .

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. \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.

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 .

5 In a geometric progression, the sum to infinity is four times the first term.

1. Show that the common ratio is \(\frac { 3 } { 4 }\).
2. Given that the third term is 9 , find the first term.
3. Find the sum of the first twenty terms.

6 A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(u _ { n } = 85 - 5 n\) for \(n \geqslant 1\).

1. Write down the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
2. Find \(\sum _ { n = 1 } ^ { 20 } u _ { n }\).
3. Given that \(u _ { 1 } , u _ { 5 }\) and \(u _ { p }\) are, respectively, the first, second and third terms of a geometric progression, find the value of \(p\).
4. Find the sum to infinity of the geometric progression in part (iii).

6

1. The first three terms of an arithmetic progression are \(2 x , x + 4\) and \(2 x - 7\) respectively. Find the value of \(x\).
2. The first three terms of another sequence are also \(2 x , x + 4\) and \(2 x - 7\) respectively.
   1. Verify that when \(x = 8\) the terms form a geometric progression and find the sum to infinity in this case.
   2. Find the other possible value of \(x\) that also gives a geometric progression.

8 \begin{figure}[h] \end{figure} Fig. 1 shows a sector \(A O B\) of a circle, centre \(O\) and radius \(O A\). The angle \(A O B\) is 1.2 radians and the area of the sector is \(60 \mathrm {~cm} ^ { 2 }\).

1. Find the perimeter of the sector. A pattern on a T-shirt, the start of which is shown in Fig. 2, consists of a sequence of similar sectors. The first sector in the pattern is sector \(A O B\) from Fig. 1, and the area of each successive sector is \(\frac { 3 } { 5 }\) of the area of the previous one. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{3836b0e7-95e6-4634-bb1e-c99b7ae3c8ba-3_362_1011_1263_568} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure}
2. (a) Find the area of the fifth sector in the pattern.
   (b) Find the total area of the first ten sectors in the pattern.
   (c) Explain why the total area will never exceed a certain limit, no matter how many sectors are used, and state the value of this limit.

9 A geometric progression has first term \(a\) and common ratio \(r\), and the terms are all different. The first, second and fourth terms of the geometric progression form the first three terms of an arithmetic progression.

1. Show that \(r ^ { 3 } - 2 r + 1 = 0\).
2. Given that the geometric progression converges, find the exact value of \(r\).
3. Given also that the sum to infinity of this geometric progression is \(3 + \sqrt { 5 }\), find the value of the integer \(a\).

7

1. The first term of a geometric progression is 7 and the common ratio is - 2 .
   1. Find the ninth term.
   2. Find the sum of the first 15 terms.
2. The first term of an arithmetic progression is 7 and the common difference is - 2 . The sum of the first \(N\) terms is - 2900 . Find the value of \(N\).

9

1. An arithmetic progression has first term \(\log _ { 2 } 27\) and common difference \(\log _ { 2 } x\).
   1. Show that the fourth term can be written as \(\log _ { 2 } \left( 27 x ^ { 3 } \right)\).
   2. Given that the fourth term is 6, find the exact value of \(x\).
2. A geometric progression has first term \(\log _ { 2 } 27\) and common ratio \(\log _ { 2 } y\).
   1. Find the set of values of \(y\) for which the geometric progression has a sum to infinity.
   2. Find the exact value of \(y\) for which the sum to infinity of the geometric progression is 3 . \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}

6 Sarah is carrying out a series of experiments which involve using increasing amounts of a chemical. In the first experiment she uses 6 g of the chemical and in the second experiment she uses 7.8 g of the chemical.

1. Given that the amounts of the chemical used form an arithmetic progression, find the total amount of chemical used in the first 30 experiments.
2. Instead it is given that the amounts of the chemical used form a geometric progression. Sarah has a total of 1800 g of the chemical available. Show that \(N\), the greatest number of experiments possible, satisfies the inequality $$1.3 ^ { N } \leqslant 91 ,$$ and use logarithms to calculate the value of \(N\).

8

1. The first term of a geometric progression is 50 and the common ratio is 0.8 . Use logarithms to find the smallest value of \(k\) such that the value of the \(k\) th term is less than 0.15 .
2. In a different geometric progression, the second term is - 3 and the sum to infinity is 4 . Show that there is only one possible value of the common ratio and hence find the first term. \section*{Question 9 begins on page 4.}

1 A geometric progression has first term 3 and second term - 6 .

1. State the value of the common ratio.
2. Find the value of the eleventh term.
3. Find the sum of the first twenty terms.

6 An arithmetic progression \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(u _ { 1 } = 5\) and \(u _ { n + 1 } = u _ { n } + 1.5\) for \(n \geqslant 1\).

1. Given that \(u _ { k } = 140\), find the value of \(k\). A geometric progression \(w _ { 1 } , w _ { 2 } , w _ { 3 } , \ldots\) is defined by \(w _ { n } = 120 \times ( 0.9 ) ^ { n - 1 }\) for \(n \geqslant 1\).
2. Find the sum of the first 16 terms of this geometric progression, giving your answer correct to 3 significant figures.
3. Use an algebraic method to find the smallest value of \(N\) such that \(\sum _ { n = 1 } ^ { N } u _ { n } > \sum _ { n = 1 } ^ { \infty } w _ { n }\).

8 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 The second term of a geometric sequence is 6 and the fifth term is - 48 .
Find the tenth term of the sequence.
Find also, in simplified form, an expression for the sum of the first \(n\) terms of this sequence.

9 A geometric progression has a positive common ratio. Its first three terms are 32, \(b\) and 12.5.
Find the value of \(b\) and find also the sum of the first 15 terms of the progression.

12 Jim and Mary are each planning monthly repayments for money they want to borrow.

1. Jim's first payment is \(\pounds 500\), and he plans to pay \(\pounds 10\) less each month, so that his second payment is \(\pounds 490\), his third is \(\pounds 480\), and so on.
   (A) Calculate his 12th payment.
   (B) He plans to make 24 payments altogether. Show that he pays \(\pounds 9240\) in total.
2. Mary's first payment is \(\pounds 460\) and she plans to pay \(2 \%\) less each month than the previous month, so that her second payment is \(\pounds 450.80\), her third is \(\pounds 441.784\), and so on.
   (A) Calculate her 12th payment.
   (B) Show that Jim's 20th payment is less than Mary's 20th payment but that his 19th is not less than her 19th.
   (C) Mary plans to make 24 payments altogether. Calculate how much she pays in total.
   (D) How much would Mary's first payment need to be if she wishes to pay \(2 \%\) less each month as before, but to pay the same in total as Jim, \(\pounds 9240\), over the 24 months?

11 A geometric progression has first term \(a\) and common ratio \(r\). The second term is 6 and the sum to infinity is 25 .

1. Write down two equations in \(a\) and \(r\). Show that one possible value of \(a\) is 10 and find the other possible value of \(a\). Write down the corresponding values of \(r\).
2. Show that the ratio of the \(n\)th terms of the two geometric progressions found in part (i) can be written as \(2 ^ { n - 2 } : 3 ^ { n - 2 }\).

11 Jill has 3 daughters and no sons. They are generation 1 of Jill's descendants.
Each of her daughters has 3 daughters and no sons. Jill's 9 granddaughters are generation 2 of her descendants. Each of her granddaughters has 3 daughters and no sons; they are descendant generation 3. Jill decides to investigate what would happen if this pattern continues, with each descendant having 3 daughters and no sons.

1. How many of Jill's descendants would there be in generation 8 ?
2. How many of Jill's descendants would there be altogether in the first 15 generations?
3. After \(n\) generations, Jill would have over a million descendants altogether. Show that \(n\) satisfies the inequality $$n > \frac { \log _ { 10 } 2000003 } { \log _ { 10 } 3 } - 1 .$$ Hence find the least possible value of \(n\).
4. How many fewer descendants would Jill have altogether in 15 generations if instead of having 3 daughters, she and each subsequent descendant has 2 daughters? \section*{END OF QUESTION PAPER}