CIE A level -Pure Mathematics 1 : Topic : 1.6 Series: arithmetic and geometric progressions : Exam Questions Paper 1

Question

The equation of a curve is $y=x^{3}+px^{2}$ ,where p is a positive constant .

(i)Show that the origin is a stationary point on the curve and find the coordinates of the other stationary point in terms of p.

(ii)Find the nature of each of the stationary points.

Another curve has equation $y=x^{3}+px^{2}+px$

(iii)Find the set of values of p for which this curve has n o stationary points.

Answer/Explanation

(i)$\frac{\mathrm{d} y}{\mathrm{d} x}=3x^{2}+2px$

Sets to 0$\rightarrow x=0$ or $-\frac{2p}{3}$

→(0,0) or$(\frac{2p}{3},\frac{4p^{3}}{27})$

(ii)$\frac{\mathrm{d} ^{2}y}{\mathrm{d} x^{2}}=6x+2p$

At (0,0)→2p positive Minimum

At$(\frac{2p}{3},\frac{4p^{3}}{27})$→-2p -ve Maximum

(iii)$y=x^{3}+px^{2}+px\rightarrow 3x^{2}+2px+p=0$

Uses $b^{2}-4ac$

$\rightarrow 4p^{2}-12p< 0$

$\rightarrow 0< p< 3$

Question

An arithmetic progression has first term a and common difference d.It is given that the sum of the first 200 terms is 4 times the sum of first 100 terms.

(i)Find d in terms of a.

(ii) Find the 100th terms of a.

Answer/Explanation

(i) $\frac{200}{2}\left ( 2a+199d \right )=4\times \frac{100}{2}\left ( 2a+99d \right )$

d=2a

(ii)a$+99d=a+99\times 2a$

199a

Question

The third term of a geometric progression is -108 and the sixth term is 32.Find

(i)the common ratio.

(ii)the first term.

(iii)the sum to infinity.

Answer/Explanation

(i)$ar^{2}=-108,ar^{5}=32$

$r^{3}=-\frac{32}{108}=\left (- \frac{8}{27} \right )$

$r=\left ( -\frac{2}{3} \right )$ or -0.666 or -0.667

(ii)a=-243

(iii)$S_{\infty }=\frac{-243}{1+\frac{2}{3}}=-\frac{729}{5}or -145.8$

Question

(i)In the expression$\left ( 1-px \right )^{6}$ ,p is a non-zero constant.Find the first three terms when$\left ( 1-px \right )^{6}$ is expanded in ascending powers of x.

(ii)It is given that the coefficient of $x^{2}$ in the expansion of $\left ( 1-x \right )\left ( 1-px \right )^{6}$

is zero .Find the value of p.

Answer/Explanation

(i) $1-6px+15p^{2}x^{2}$

(ii)$15p^{2}\times 1-6p\times -1$

$3p\left ( 5p+2 \right )=0$

$p=-\frac{2}{5}$

Question

(a)The first two terms of an arithmetic progression are 1 and $\cos ^{2}x$ respectively. Show that the sum of the first ten terms can be expressed in the form $a-b\sin ^{2}x$,where a and b are constants to be found.

(b)The first two terms of a geometric progression are 1 and $\frac{1}{3}\tan ^{2}\Theta $ respectively, where $0< \Theta < \frac{1}{2}\pi $

(i) Find the set of values of $\Theta$ for which the progression is convergent.

(ii)Find the exact value of the sum to infinity when $\Theta =\frac{1}{6}\pi $

Answer/Explanation

(a) $S_{10}=\frac{10}{2\left [ 2+9\left ( \cos ^{2}x-1 \right ) \right ]}$

$S_{10}=5\left [ 2-9\sin ^{2}x \right ]$

$S_{10}=10-45\sin ^{2}x$

(b)(i)$0< \frac{1}{3}\tan ^{2}\Theta < 1$

$0< \Theta < \frac{\pi }{3}$

(ii)$S_{\infty }=\frac{1}{1-\frac{1}{3}\tan ^{2}\frac{\pi }{6}}$

$S_{\infty }=\frac{9}{8}$ or 1.125

Question

(a)A geometric has first 3a and common ratio r. A second geometric progression has first term a and common ratio -2r.The two progressions have the same sum to infinity. Find the value of r.

(b) The first two terms of an arithmetic progression are 15 and 19 respectively. The first two terms of a second arithmetic progression are 420 and 415 respectively.The two progression have sum of first n terms. Find the value of n.

Answer/Explanation

(i)$\frac{3a}{1-r}=\frac{a}{1+2r}$

$3+6r=1-r$

$r=-\frac{2}{7}$

(ii)$\frac{1}{2}\left [ 2\times 15+\left ( n-1 \right ) 4\right ]=\frac{1}{2}n\left [ 2\times 420+(n-1)\left ( -5 \right ) \right ]$

n=91

Question

(i)Find the first 3 terms in the expansion of $\left ( 2x-x^{2} \right )^{6}$ in ascending powers of x.

(ii)Hence find the coefficient of $x^{8}$ in the expansion of $\left ( 2+x \right )\left ( 2x-x^{2} \right )^{6}$.

Answer/Explanation

(i)$(2x-x^{2})^{6}=64x^{6}-192x^{7}+240x^{8}$

(ii)coefficient of $x^{8}$=2\times 240-192

288

Question

The first term of an arithmetic progression is 61 and the second term is 57.The sum of the first n terms is n.Find the value of the positive integers n.

Answer/Explanation

$\frac{n}{2}\left [ 122+\left ( n-1 \right )(-4) \right ]$

$n=\frac{n}{2}\left [ 122+\left ( n-1 \right )(-4) \right ]$

2n(n-31)=0

n=31

Question

The first term of a progression is 4x and the second term is $x^{2}$
(i) For the case where the progression is arithmetic with a common difference of 12, find the possible values of x and the corresponding values of the third term.
(ii) For the case where the progression is geometric with a sum to infinity of 8, find the third term.

Answer/Explanation

(i)$x^{2}-4x=12$

x=-2 or 6

Third term$=(-2)^{2}+12=16$ or $6^{2}+12=48$

(ii)$r^{2}=\frac{x^{2}}{4x}=\frac{x}{4}$

$\frac{4x}{1-\frac{x}{4}}=8$

$x=\frac{4}{3}$ or $r=\frac{1}{3}$

ALT

$\frac{4x}{1-r}=8\rightarrow r=1-\frac{1}{2}x$ or $\frac{4x}{1-r}=8\rightarrow x=2(1-r)$

$x^{2}=4x(1-\frac{1}{2}x)$ $r=\frac{2(1-r)}{4}$

$x=\frac{4}{3}$ $r=\frac{1}{3}$

Question

(a) In an arithmetic progression the sum of the first ten terms is 400 and the sum of the next ten terms is 1000. Find the common difference and the first term.
(b) A geometric progression has first term a, common ratio r and sum to infinity 6. A second geometric progression has first term 2a, common ratio $r^{2}$ and sum to infinity 7. Find the values of a and r.

Answer/Explanation

(a)$\frac{10}{2}(2a+9d)=400$

$\frac{20}{2}(2a+19d)=1400$

$\frac{10}{2}\left [ 2(a+10d)+9d \right ]=1000$

d=6 a=13

(b)$\frac{a}{1-r}=6 $ $\frac{2a}{1-r^{2}}=7$

$\frac{12(1-r)}{1-r^{2}}=7$ or $\frac{1-r^{2}}{1-r}=\frac{12}{7}$

$r=\frac{5}{7}$ or 0.714

$a=\frac{12}{7}$ or 1.714

Question

(i) A geometric progression has first term a (a ≠ 0), common ratio r and sum to infinity S. A second geometric progression has first term a, common ratio 2r and sum to infinity 3S. Find the value of r.
(ii) An arithmetic progression has first term 7. The nth term is 84 and the (3n)th term is 245. Find the value of n.

Answer/Explanation

(i)$S=\frac{a}{1-r}$ ,$3S=\frac{a}{1-2r}$

1-r=3-6r

$r=\frac{2}{5}$

(ii)$7+(n-1)d=84$ and/or $7+(3n-1)d=245$

$\left [ (n-1)d=77,(3n-1)d=238,2nd=161 \right ]$

$\frac{n-1}{3n-1}=\frac{77}{238}$ (must be from the correct $u_{n}$ formula)

n=23 $(d=\frac{77}{22}=3.5)$

Question.

(a) An arithmetic progression has a first term of 32, a 5th term of 22 and a last term of −28. Find the
sum of all the terms in the progression.

(b) Each year a school allocates a sum of money for the library. The amount allocated each year
increases by 2.5% of the amount allocated the previous year. In 2005 the school allocated $$2000$.
Find the total amount allocated in the years 2005 to 2014 inclusive.

Answer/Explanation

(a) $a=32,a+4d=22\rightarrow d=-2.5$
$a+(n-1)d=-28\rightarrow n=25$

$S_{25}=\frac{25}{2}(64-2.5\times 24)=50$

(b)a=2000,r=1.025

$S_{10}=2000\left ( \frac{1.025^{10}-1}{1.025-1} \right )=22400$ or a value which rounds to this

Question.

(a) The first term of a geometric progression in which all the terms are positive is 50. The third term is 32. Find the sum to infinity of the progression.

(b) The first three terms of an arithmetic progression are 2 sin x, 3 cos x and (sin x + 2 cos x) respectively, where x is an acute angle.
(i) Show that $tan x = \frac {4}{3}$
(ii) Find the sum of the first twenty terms of the progression.

Answer/Explanation

a=50,$ar^{2}=32$

$\rightarrow r=\frac{4}{5}$ (allow $\frac{4}{5}$ for M mark)

$\rightarrow S_{\infty }=250$

(b) (i) $2\sin x,3\cos x,\left ( \sin x+2\cos x \right )$

3c-2s=(s+2c)-3c

(or uses a,a+d,a+2d)

$\rightarrow 4c=3s\rightarrow t=\frac{4}{3}$

SC uses $\rightarrow 4c=3s\rightarrow t=\frac{4}{3}$

$u_{1}=\frac{8}{5},u_{2}=\frac{9}{5},u_{3}=\frac{10}{5}$

(ii)$\rightarrow c=\frac{3}{5},s=\frac{4}{5}$ or calculator $ x=53.1^{\circ}$

→a=1.6,d=0.2

$S_{\infty }=70$

Question.

The function f is defined for x≥ 0 by $f(x) =(4x + 1)^\frac{3}{2}$.

(i) Find f′(x) and f′′(x).

The first, second and third terms of a geometric progression are respectively f(2), f′(2) and kf′′(2).
(ii) Find the value of the constant k.

Answer/Explanation

(i) $f{}’\left ( x \right )=\left [ \frac{3}{2}\left ( 4x+1 \right )^{\frac{1}{2}} \right ]\left [ 4 \right ]$

$f{}”\left ( x \right )=6\times \frac{1}{2}\times \left ( 4x+1 \right )^{-\frac{1}{2}}\times 4$

(ii) f(2),$f{}’\left ( 2 \right )$,$kf{}”\left ( 2 \right )=27$,4k OR 12

$\frac{27}{18}=\frac{18}{4k}$ oe OR $kf{}”\left ( 2 \right )=12\Rightarrow k=3$

Question.

The 12th term of an arithmetic progression is 17 and the sum of the first 31 terms is 1023. Find the 31st term.

Answer/Explanation

Ans: $a+11d=17$
$\frac{31}{2}(2a+30d)=1023$
on solving both ,
$d=4$ , $a=-27$
31st term $=93$

Question

(a) The third and fourth terms of a geometric progression are 48 and 32 respectively. Find the sum
to infinity of the progression.

(b) Two schemes are proposed for increasing the amount of household waste that is recycled each
week.

Scheme A is to increase the amount of waste recycled each month by 0.16 tonnes.

Scheme B is to increase the amount of waste recycled each month by 6% of the amount recycled in the previous month.

The proposal is to operate the scheme for a period of 24 months. The amount recycled in the
first month is 2.5 tonnes.For each scheme, find the total amount of waste that would be recycled over the 24- month period.

Answer/Explanation

Queastion

(i) The first and second terms of a geometric progression are p and 2p respectively, where p is a
positive constant. The sum of the first n terms is greater than 1000p. Show that 2n > 1001.

(ii) In another case, p and 2p are the first and second terms respectively of an arithmetic progression.
The nth term is 336 and the sum of the first n terms is 7224. Write down two equations in n and
p and hence find the values of n and p.

Answer/Explanation

(i)

$S_{n}=\frac{p(2^{n}-1)}{2-1} $

$p(2^{n}-1)> 1000p\rightarrow 2^{n}> 1001$

(ii)

$ p + ( n−1)=336$

$\frac{n}{2}\left [ 2p=(n-1) p\right ]=7224$

Eliminate n or p to an equation in one variable $n = 42$ ,
$p = 8$

Question

A geometric progression has first term a, common ratio r and sum to infinity S. A second geometric
progression has first term a, common ratio R and sum to infinity 2S.
(a) Show that r = 2R − 1. [3]

It is now given that the 3rd term of the first progression is equal to the 2nd term of the second
progression.
(b) Express S in terms of a. [4]

Answer/Explanation

Ans

8 (a) $S=\frac{1}{1-r}, \ 2S=\frac{a}{1-R}$

$\frac{2a}{1-r}=\frac{a}{1-R}$

$2-2R=1-r\rightarrow r=2R-1$

8 (b) $ar^{2}=aR\rightarrow (a)(2R-1)^{2}=R(a)$

$4R^{2}-5R+1(=0\rightarrow )(4R-1)(R-1)(=0)$

$R=\frac{1}{4}$

$S=\frac{2a}{3}$

Alternative method for question 8(b)

$ar^{2}=aR\rightarrow (a)r^{2}=\frac{1}{2}(r+1)(a)$

$2r^{2}-r-1(=0)\rightarrow (2r+1)(r-1)(=0)$

$r=-\frac{1}{2}$

$S=\frac{2a}{3}$

Question

A woman’s basic salary for her first year with a particular company is $30000 and at the end of the year she also gets a bonus of $600.
(a) For her first year, express her bonus as a percentage of her basic salary.
At the end of each complete year, the woman’s basic salary will increase by 3% and her bonus will increase by $100.
(b) Express the bonus she will be paid at the end of her 24th year as a percentage of the basic salary paid during that year.

Answer/Explanation

Ans:

(a) 2%
(b) Bonus $=600+23 \times 100 = 2900$
Salary $=30000 \times 1.03^{23}$
= 59207.60
$\frac{their2900}{their59200}$
4.9(0)%

Question

The sum of the first 20 terms of an arithmetic progression is 405 and the sum of the first 40 terms
is 1410.
Find the 60th term of the progression. [5]

Answer/Explanation

Ans

2 10(2a + 19d) = 405
20(2a + 39d) = 1410
Solving simultaneously two equations obtained from using the correct sum
formulae [a = 6, d = 1.5]
Using the correct formula for 60th term with their a and d
60th term = 94.5

Question

The first term of a geometric progression and the first term of an arithmetic progression are both equal to a.
The third term of the geometric progression is equal to the second term of the arithmetic progression.
The fifth term of the geometric progression is equal to the sixth term of the arithmetic progression.
Given that the terms are all positive and not all equal, find the sum of the first twenty terms of the arithmetic progression in terms of a.

Answer/Explanation

Ans:

$ar^{2} = a + d$
$ar^{4} = a + 5d$

$a^{2} r^{4} = a (a+5d) leading to a^{2} + 5ad = (a+d)^{2}$

$[3ad -d^{2} = 0 leading] d = 3a OR [r= 2 leading to] d = 3a$

$S_{20} = \frac{20}{2} [2a + 19 \times 3a]$

590a

Question

The first term of a progression is $cos\theta$, where $0<\theta<\frac{1}{2}\pi$.
(a) For the case where the progression is geometric, the sum to infinity is $\frac{1}{cos\theta}$.
(i) Show that the second term is $cos\theta sin^2 \theta$.
(ii) Find the sum of the first 12 terms when $\theta=\frac{1}{3}\pi$, giving your answer correct to 4 significant figures.
(b) For the case where the progression is arithmetic, the first two terms are again $cos \theta $ and $cos \theta sin^2 \theta$ respectively.
Find the 85th term when $\theta =\frac{1}{3}\pi$.

Answer/Explanation

Ans:

(a) (i) $\frac{cos \theta}{1-r}=\frac{1}{cos \theta}$
$1-r=cos^2 \theta$ leading to $r=1-cos^2 \theta$
$r=sin^2 \theta$ leading to 2nd term $=cos \theta sin^2 \theta$
(a)(ii) $S_{12}=\frac{cos(\frac{\pi}{3})[1-(sin^2(\frac{\pi}{3}))^{12}]}{1-sin^2(\frac{\pi}{3})}$=$0.5[1-(0.75)^{12}]}{1-0.75}$
1.937
(b) $[d=]cos\theta sin^2 \theta – cos \theta$
$-\frac{1}{8}$
[85th term=] $\frac{1}{2} + 84 \times – \frac{1}{8}$
-10

Question

The equation of a curve is $y=2\sqrt{3x+4}-x$

(a) Find the equation of the normal to the curve at the point (4, 4), giving your answer in the form
y = mx + c. [5]
(b) Find the coordinates of the stationary point. [3]

(d) Find the exact area of the region bounded by the curve, the x-axis and the lines x = 0 and x = 4. [4]

Answer/Explanation

Ans

11 (a) $\frac{dx}{dy}=3(3x+4)^{-0.5}-1$

Gradient of tangent $=-\frac{1}{4}$ and Gradient of normal = 4

Equation of line is (y – 4) = 4(x – 4) or evaluate c

So y = 4x – 12

11 (b) 3(3x+4)^-0.5 -1 = 0

Solving as far as x=

$x=\frac{5}{3}, y=2\left ( 3\times \frac{5}{3}+4 \right )^{0.5}-\frac{5}{3}=\frac{13}{3}$

At $x=\frac{5}{3} \frac{d^{2}y}{dx^{2}}$ is negative so the point is a maximum

11 (d) $Area-[f2(3x+4)^{0.5}-xdx=]\frac{4}{9}(3x+4)^{1.5}-\frac{1}{2}x^{2}$

$\left ( \frac{4}{9}(16)^{1.5}-\frac{1}{2}(4)^{2} \right )-\frac{4}{9}(4)^{1.5}=\frac{256}{9}-8-\frac{32}{9}$

$16\frac{8}{9}$

Question.

Each year the selling price of a diamond necklace increases by 5% of the price the year before. The selling price of the necklace in the year 2000 was $$36 000$.
(a) Write down an expression for the selling price of the necklace n years later and hence find the
selling price in 2008.

(b) The company that makes the necklace only sells one each year. Find the total amount of money obtained in the ten-year period starting in the year 2000.

Answer/Explanation

(a) $$36 000 \times (1.05)^{n}$
(B1 for r = 1.05. M1 method for rth term)

$ $53 200$ after 8 years.

(b) $S_{10}=36000\frac{(1.05^{10}-1)}{(1.05-1)}$

$ $453 000 $

Question.

The first term of an arithmetic progression is a and the common difference is −4. The first term of a geometric progression is 5a and the common ratio is $-\frac{1}{4}$. The sum to infinity of the geometric progression is equal to the sum of the first eight terms of the arithmetic progression.

(a) Find the value of a.

The kth term of the arithmetic progression is zero.

(b) Find the value of k

Answer/Explanation

(a) $\frac{5a}{1-\left ( \pm \frac{1}{4} \right )}$

$\frac{8}{2}\left [ 2a + 7(-4) \right ]$

4a = 8a – 112 leading to a = [28]

a = 28

(b) their k

28 + ( k – 1) (-4_) = 0

[k = ] 8