Archive for March, 2017

DSE Math 2016 Paper 1 ANS

Posted: March 27, 2017 in Mathematics

Exam week’s coming, so here it is.

Section A (35 marks)

(1)  Simplify \displaystyle\frac{(x^8 y^7)^2}{x^5 y^{-6}} and express your answer with positive indices.  (3 marks)

Solution

\displaystyle\frac{(x^8 y^7)^2}{x^5 y^{-6}} = \frac{x^{16}y^{14}}{x^5 y^{-6}} = x^{11}y^{20}


(2)  Make x the subject of the formula Ax=(4x+B)C.  (3 marks)

Solution

Ax=4Cx+BC

Ax-4Cx=BC

\therefore x = \dfrac{BC}{A-4C}


(3)  Simplify \dfrac{2}{4x-5}+\dfrac{3}{1-6x}.  (3 marks)

Solution

\displaystyle \frac{2(1-6x)+3(4x-5)}{(4x-5)(1-6x)} = \frac{2-12x+12x-15}{(4x-5)(1-6x)} = \frac{13}{(4x-5)(6x-1)}


(4)  Factorize

(a)  5m-10n.

(b)  m^2+mn-6n^2.

(c)  m^2+mn-6n^2-5m+10n.  (4 marks)

Solution

(a)  5(m-2n)

(b)  (m+3n)(m-2n)

(c)  (m+3n)(m-2n)-5(m-2n) = (m-2n)(m+3n-5)


(5)  In a recreation club, there are 180 members and the number of male members is 40% more than the number of female members. Find the difference of the number of male members and the number of female members.  (4 marks)

Solution

Let f be the number of female members.

1.4f + f = 180 \quad \Rightarrow \quad f = \dfrac{180}{2.4} = 75

\therefore \text{difference} = 1.4f-f=0.4f=0.4(75)=30


(6)  Consider the compound inequality

x+6 < 6(x+11) \text{ or } x\leq -5 \qquad (*)

(a)  Solve (*).

(b)  Write down the greatest negative integer satisfying (*).  (4 marks)

Solution

(a)  x+6<6x+66 \text{ or } x\leq -5 \\ \Leftrightarrow 5x>-60 \text{ or } x\leq -5 \\ \Leftrightarrow x>-12 \text{ or } x\leq -5 \\ \Leftrightarrow x\in\mathbb{R}

(b)  -1


(7)  In a polar coordinate system, O is the pole. The polar coordinate of the points A and B are (12, 75^\circ) and (12, 135^\circ) respectively.

(a)  Find \angle AOB.

(b)  Find the perimeter of \triangle AOB.

(c)  Write down the number of folds of rotational symmetry of \triangle AOB.  (4 marks)

Solution

(a)  \angle AOB = 135^\circ - 75^\circ = 60^\circScreenshot_2017-03-25_04-49-31

(b)  Obviously the triangle is isosceles and hence \alpha = \beta = 60^\circ and hence \triangle AOB is equilateral. Therefore the perimeter is 36.

(c)  3


(8)  It is given that f(x) is the sum of two parts, one part varies as x and the other part varies as x^2. Suppose that f(3)=48 and f(9)=198.

(a)  Find f(x).

(b)  Solve the equation f(x)=90.  (5 marks)

Solution

(a)  Let f(x) = ax^2+bx for some non-zero real numbers a and b.

f(3) = 9a+3b = 48 \Leftrightarrow 3a+b=16 \qquad (1)

f(9) = 81a+9b = 198 \Leftrightarrow 9a+b=22 \qquad (2)

(2) – (1) gives 6a=6 \Rightarrow a=1 \Rightarrow b=13

\therefore f(x) = x^2+13x

(b)  x^2+13x-90=0 \Rightarrow (x+18)(x-5)=0 \Rightarrow x=-18 \text{ or } x=5


(9)  The frequency distribution table and the cumulative frequency distribution table below show the distribution of the heights of the plants in a garden.

Screenshot_2017-03-25_05-37-35.png

(a)  Find x, y, and z.

(b)  If a plant is randomly selected from the garden, find the probability that the height of the selected plant is less than 1.25 m but not less than 0.65 m.

Solution

(a)  a=2, \therefore x=2+4=6, y=37-15=22, z=37+3=40

(b)  \dfrac{22-6}{40}=\dfrac{16}{40}=\dfrac{2}{5}


(10)  The coordinates of the points A and B are (5, 7) and (13,1) respectively. Let P be a moving point in the rectangular coordinate plane such that P is equidistant from A and B. Denote the locus of P by \Gamma.

(a)  Find the equation of \Gamma.  (2 marks)

(b)  \Gamma intersects the x-axis and y-axis at H and K respectively. Denote the origin by O. Let C be the circle which passes through O, H, and K. Someone claims that the circumference of C exceeds 30. Is the claim correct? Explain your answer.  (3 marks)

Solution

(a)  (x-5)^2+(y-7)^2=(x-13)^2+(y-1)^2

\Rightarrow -10x+25-14y+49=-26x+169-2y+1

\Rightarrow 16x=12y+96

\Rightarrow 4x-3y-24=0

(b)  Let y=0, then x=6, hence H(6, 0)

Let x=0, then y=-8, hence K(0, -8)

Since \triangle HKO is a right triangle where \angle O = 90^\circ, hence HK is the diameter of the circle C.

\text{Circumference} = \pi d = \pi\sqrt{6^2+8^2} = 10\pi > 30

Therefore, the claim is correct.


(11)  An inverted right circular conical vessel contains some milk. The vessel is held vertically. The depth of milk in the vessel is 12 \text{ cm}. Peter pours 444\pi\text{ cm}^2 of milk into the vessel without overflowing. He now finds that the depth of milk in the vessel is 16\text{ cm}.

(a)  Express the final volume of milk in the vessel in terms of \pi.  (3 marks)

(b)  Peter claims that the final area of the wet curved surface of the vessel is at least 800\text{ cm}^2. Do you agree? Explain your answer.  (3 marks)

Solution

(a)  Let x be the original volume of milk.

\displaystyle\frac{444\pi+x}{x}=\left(\frac{16}{12}\right)^3=\left(\frac{4}{3}\right)^3

27(444\pi+x)=64x

27\cdot444\pi=37x

x=27\cdot4\cdot3\pi = 81\cdot4\pi=324\pi

\therefore \text{Final volume} = 444\pi+324\pi = 768\pi\text{ cm}^3

(b)  \dfrac{\pi r^2 (16)}{3} = 768 \pi \Rightarrow r^2=144 \Rightarrow r=12\text{ cm}

\text{Curved surface area} = \pi r s = \pi(12)\sqrt{12^2+16^2} = 240\pi \approx 754 \text{ cm}^2 < 800\text{ cm}^2

Therefore, disagree.


(12)  The bar chart below shows the distribution of the ages of the children in a group, where a>11 and 4<b<10. The median of the ages of the children in the group is 7.5.

bar-chart.png

(a)  Find a and b.  (3 marks)

(b)  Four more children now join the group. It is found that the ages of these four children are all different and the range of the ages of the children in the group remains unchanged. Find

(i)  the greatest possible median of the ages of the children in the group.

(ii)  the least possible mean of the ages of the children in the group.  (4 marks)

Solution

(a)  Since the median age is 7.5, hence 11+a=11+b+4 \Rightarrow a=b+4

Now, a>11 \text{ and } 4<b<10 \Rightarrow b>7 \text{ and } 4<b<10.

Since a,b\in\mathbb{Z}^+ \therefore b=8\text{ or } 9

\therefore (a,b) = (12,8)\text{ or }(13,9)

(b) (i)  To maximize the median, we add age 7, 8, 9, 10. Therefore the new median will be 8.

(b) (ii)  To minimize the mean, we add age 6, 7, 8, 9.

Case (a, b)=(12,8),

\text{mean} = \dfrac{12\cdot6+13\cdot7+12\cdot8+9\cdot9+4\cdot10}{12+13+12+9+4} = \dfrac{380}{50}=\dfrac{38}{5}=7.6

Case (a, b)=(13,9),

\text{mean} = \dfrac{12\cdot6+14\cdot7+12\cdot8+10\cdot9+4\cdot10}{12+14+12+10+4} = \dfrac{396}{52}=\dfrac{99}{13}=7.\overline{615384}


(13)  In figure 1, ABC is a triangle. D, E, and M are points lying on BC such that BD=CE, \angle ADC = \angle AEB and DM=EM.

Screenshot_2017-03-25_18-17-40

(a)  Prove that \triangle ACD\cong\triangle ABE.  (2 marks)

(b) Suppose that AD=15\text{ cm}, BD=7\text{ cm} and DE=18\text{ cm}.

(i)  Find AM.

(ii)  Is \triangle ABE a right-angled triangle? Explain your answer.  (5 marks)

Solution

(a)  \angle ADC = \angle AEB \quad \text{(given)}

AD=AE \quad \text{(sides opp. equal }\angle\text{s)}

BD=CE \quad\text{(given)}

BE = BD+DE = CE+DE = CD\quad (BD=CE)

\therefore \triangle ACD\cong\triangle ABE \quad\text{(SAS)}

(b) (i)  AD=AE=15 cm\quad \text{(given)}

Hence, \triangle ADE is isosceles.

DM=EM\quad\text{(given)}

\therefore AM \perp DE\quad\text{(property of isos }\triangle \text{)}

\therefore AM=\sqrt{15^2-9^2} = 12\text{ cm} \quad\text{(Pyth. thm)}

(b) (ii) DM=EM=9\text{ cm}

By Pyth. thm, AB=\sqrt{12^2+(7+9)^2} = 20\text{ cm}

AB^2+AE^2 = 20^2+15^2 = 25^2 = AE^2

Therefore, \triangle ABE is right-angled.


(14)  Let p(x)=6x^4+7x^3+ax^2+bx+c, where a, b, and c are constants. When p(x) is divided by x+2 and when p(x) is divided by x-2, the two remainders are equal. It is given that p(x)=(lx^2+5x+8)(2x^2+mx+n), where l, m, and n are constants.

(a)  Find l, m, and n.  (5 marks)

(b)  How many real roots does the equation p(x)=0 have? Explain your answer.  (5 marks)

Solution

(a)  p(-2)=p(2)

6(-2)^4+7(-2)^3+a(-2)^2+b(-2)+c = 6(2)^4+7(2)^3+a(2)^2+b(2)+c

-56-2b = 56+2b

4b = -112

b=-28

\therefore p(x)=6x^4+7x^3+ax^2-28x+c

(lx^2+5x+8)(2x^2+mx+n) = (2l)x^4+(lm+10)x^3+(ln+5m+16)x^2+(5n+8m)x+8n

By matching the coefficient,

2l=6 \Rightarrow l=3

lm+10=7 \Rightarrow m=-1

5n+8m=-28 \Rightarrow n=-4

(b)  p(x)=(lx^2+5x+8)(2x^2+mx+n)=(3x^2+5x+8)(2x^2-x-4)=0

\therefore (3x^2+5x+8)=0\text{ or }(2x^2-x-4)=0

For 3x^2+5x+8=0, \quad \Delta=25-4\cdot3\cdot8 = -71\quad \therefore no real root.

For 2x^2-x-4=0, \quad \Delta=1+4\cdot2\cdot4 = 33\quad \therefore two real roots.

Therefore, p(x)=0 has two real roots.


Section B (35 marks)

(15)  If 4 boys and 5 girls randomly form a queue, find the probability that no boys are next to each other in the queue.  (3 marks)

Solution

Consider this : _ G _ G _ G _ G _ G _

There are 6 places to put 4 B’s so there are 6P4 ways to place the boys. Moreover, there are 5! ways to permute those 5 G’s. Therefore,

\displaystyle \text{P(no boys are next to each other)}=\frac{_6P_4 \cdot 5!}{9!} = \frac{6\cdot5\cdot4\cdot3}{9\cdot8\cdot7\cdot6} = \frac{5}{42}


(16)  In a test, the mean of the distribution of the scores of a class of students is 61 marks. The standard scores of Albert and Mary are -2.6 and 1.4 respectively. Albert gets 22 marks. A student claims that the range of the distribution is at most 59 marks. Is the claim correct? Explain your answer.  (3 marks)

Solution

Z=\dfrac{X-\mu}{\sigma} \quad \Rightarrow\quad -2.6 = \dfrac{22-61}{\sigma} \quad\Rightarrow\quad \sigma = 15

So, the standard deviation is 15. Now, let’s calculate what Mary gets on the test.

X=Z\sigma+\mu = 1.4\cdot 15 + 61 = 82

82-22 = 60, which is greater than 59. Therefore the claim is false.


(17)  The 1st term and the 38th term of an arithmetic sequence are 666 and 555 respectively. Find

(a)  the common difference of the sequence.  (2 marks)

(b)  the greatest value of n such that the sum of the first n term of the sequence is positive.  (3 marks)

Solution

(a)  t_{38} =t_1 + 37d

\therefore d=\dfrac{t_{38}-t_1}{37} = \dfrac{555-666}{37} = \dfrac{-111}{37}=-3

(b)  S_n = \dfrac{(2a+(n-1)d)n}{2} > 0

\displaystyle \frac{(2(666)+(n-1)(-3))n}{2}>0

Since n is positive integer, we can divide both side by n without switching the sign.

2(666)-3(n-1)>0

n-1 < 2(222)

n < 445

Therefore, the greatest such n is 444.


(18)  Let f(x) = \dfrac{-1}{3}x^2 + 12x - 121.

(a)  Using the method of completing the square, find the coordinates of the vertex of the graph of y = f(x).  (2 marks)

(b)  The graph of y=g(x) is obtained by translating the graph of y=f(x) vertically. If the graph of y=g(x) touches the x-axis, find g(x).  (2 marks)

(c)  Under a transformation, f(x) is changed to $latex \dfrac{-1}{3}x^2 – 12x – 121$. Describe the geometric meaning of the transformation.  (2 marks)

Solution

(a)  \displaystyle f(x) = \frac{-1}{3}x^2 + 12x - 121

= -\frac13 (x^2-36x+18^2-18^2) - 121

= -\frac13(x-18)^2+\frac{18^2}{3}-121

= -\frac13(x-18)^2 -13

Therefore, the vertex is (18, -13).

(b)  Basically translate the graph of f(x) 13 units upward.

\therefore g(x) = -\frac13(x-18)^2

(c)  Since \displaystyle\frac{-1}{3}x^2-12x-121=\frac{-1}{3}(-x)^2+12(-x)-121 = f(-x)

Therefore, the transformation is reflecting the graph of f(x) horizontally along the y-axis.


(19)  Figure 2 shows a geometric model ABCD in the form of tetrahedron. It is given that \angle BAD=86^\circ, \angle CBD=43^\circ, AB=10\text{ cm}, AC=6\text{ cm}, BC=8\text{ cm} \text{ and } BD=15\text{ cm}.

triangle.png

(a)  Find \angle ABD and CD.  (4 marks)

(b)  A craftsman claims that the angle between AB and the face BCD is \angle ABC. Do you agree? Explain your answer.  (2 marks)

Solution

(a)  By Sine Law, \displaystyle \frac{\sin\angle ADB}{AB}=\frac{\sin\angle BAD}{BD}

\displaystyle \frac{\sin\angle ADB}{10}=\frac{\sin 86^\circ}{15}

\therefore \angle ADB = \sin^{-1} \dfrac{10\sin 86^\circ}{15}

\therefore \angle ABD = 180^\circ - 86^\circ - \sin^{-1} \dfrac{2\sin 86^\circ}{3} = 52.31439868 \approx 52.3^\circ \text{ (3 s.f.)}

By Cosine Law, CD^2=AC^2+AD^2-2AC\cdot AD \cos\angle CBD.

\therefore CD = \sqrt{8^2+15^2-2\cdot8\cdot15\cdot\cos 43^\circ} = 10.65246974 \approx 10.7 \text{ cm  (3 s.f.)}

(b)  By Sine law, \displaystyle \frac{AD}{\sin\angle ABD}=\frac{BD}{\sin\angle BAD}

\therefore AD=\dfrac{15\sin\angle ABD}{\sin 86^\circ} \approx 11.8996447\ldots

AC^2+CD^2=6^2+10.65246974^2=149.4751116\neq AD^2=141.6015440

Therefore, do not agree.


(20)  \triangle OPQ is an obtuse-angled triangle. Denote the in-centre and the circumcentre of \triangle OPQ by I and J respectively. It is given that P, I, and J are collinear.

(a)  Prove that OP=PQ.  (3 marks)

(b)  A rectangular coordinate system is introduced so that the coordinates of O and Q are (0,0) and (40,30) respectively while the y-coordinate of P is 19. Let C be the circle which passes through O, P, and Q.

(i)  Find the equation of C.

(ii)  Let L_1 and L_2 be two tangents to C such that the slope of each tangent is \dfrac34 and the y-intercept of L_1 is greater than that of L_2. L_1 cuts the x-axis and y-axis at S and T respectively while L_2 cuts the x-axis and the y-axis at U and V respectively. Someone claims that the area of the trapezium STUV exceeds 17000. Is the claim correct? Explain your answer.  (9 marks)

Solution

(a)  Since P, I, and J are collinear, we can draw a line passing through all three points and meet the line segment OQ at K. Since I is incenter, PI is angle bisector of \angle OPQ, hence \angle OPK = \angle QPK. Since J is circumcenter, then PK is perpendicular bisector of OQ and so \angle PKO = \angle PKQ = 90^\circ. PK = PK obviously by common sides. Hence, \triangle PKO \cong \triangle PKQ by ASA. Therefore the corresponding sides of the two congruent triangles are congruent and so OP=PQ.

(b) (i)  Let P=(x, 19).  Since OP = PQ,

\sqrt{x^2+19^2} = \sqrt{(x-40)^2+(19-30)^2}

x^2+361=x^2-80x+1600+121

80x=1360

x=17

Let circle C: x^2+y^2+Dx+Ey+F=0

Substitute O(0,0) into circle C yields F=0. Hence,

P(17, 19): 17^2+19^2+17D+19E=0 \Rightarrow 650+17D+19E \qquad (1)

Q(40, 30): 40^2+30^2+40D+30E=0 \Rightarrow 250+4D+3E=0 \qquad (2)

3(1)-19(2):

1950+51D-4750-76D=0

\therefore D=-112

\therefore E=66

Therefore, the equation of circle C is x^2+y^2-112x+66y=0

(b) (ii)  Since L_1 and L_2 have slope of \dfrac34, They both have the equation of the form: y=\dfrac34x+b

They are both tangents to the circle hence:

x^2+y^2-112x+66y =0 \quad\text{and}\quad y=\dfrac34x+b

x^2+\left(\frac34x+b\right)^2-112x+66\left(\frac34x+b\right)=0

x^2+\frac{9}{16}x^2+\frac{3b}{2}x+b^2-112x+\frac{99}{2}x+66b=0

16x^2+9x^2+24bx+16b^2-1792x+792x+1056b=0

25x^2+(24b-1000)x+16b^2+1056b=0

Since they are tangents to the circle,

\Delta = (24b-1000)^2-4(25)(16b^2+1056b)=0

576b^2-48000b+1000000-1600b^2-105600b=0

1024b^2+153600b-1000000=0

16b^2+2400b-15625=0

(4b+625)(4b-25)=0

b=-\dfrac{625}{4} \text{ or } b=\dfrac{25}{4}

\displaystyle\therefore \quad L_1: y=\frac34x+\frac{25}{4} \quad\text{and}\quad L_2: y =\frac34x-\frac{625}{4}

L_1: Let y=0 \Rightarrow x=-\dfrac{25}{3} \Rightarrow S\left(-\dfrac{25}{3},0\right) \Rightarrow T\left(0,\dfrac{25}{4}\right)

L_2: Let y=0 \Rightarrow x=\dfrac{625}{3} \Rightarrow U\left(\dfrac{625}{3},0\right) \Rightarrow V\left(0,-\dfrac{625}{4}\right)

\displaystyle \therefore ST=\sqrt{\left(\frac{25}{3}\right)^2+\left(\frac{25}{4}\right)^2}=25\sqrt{\frac{1}{9}+\frac{1}{16}}=\frac{25}{12}

\displaystyle \therefore UV=\sqrt{\left(\frac{625}{3}\right)^2+\left(\frac{625}{4}\right)^2}=625\sqrt{\frac{1}{9}+\frac{1}{16}}=\frac{3125}{12}

C: x^2+y^2-112x+66y=0

\Rightarrow x^2-112x+56^2+y^2+66y+33^2=56^2+33^2=4225

\Rightarrow (x-56)^2+(y+33)^2=65^2 \Rightarrow r=65 \Rightarrow d=130

\displaystyle\therefore \text{Area} = \frac{(ST+UV)d}{2} = \left(\frac{125}{12}+\frac{3125}{12}\right)65 =17604.1\overline{6}>17000

Therefore, the claim is correct.

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777, 365, and 21

Posted: March 26, 2017 in Mathematics

The number “689” has been the laughingstock of the HK chief executive for the past 4 years. “689” basically means 冇柒用, so in response, this year’s chief executive will receive “777” votes just to emphasis that she is very 好有柒用. It’s not one 7 or two 7s, but three 7s in a row, how nice!

Actually 777 is quite an interesting number. In gambling, it is a lucky number to most people because “777” is a jackpot number. In computer science, “777” means you have been granted read, write, and executable permission on a file (in UNIX or UNIX-like operating system such as Linux, BSD, Mac OSX, etc). To most HKers, 777, 365, and 21 mean the number of vote of today’s HK chief executive election. To me, I have another interpretation.

777 = 7 \cdot 111 = 7 \cdot 3 \cdot 37

365 = 5 \cdot 73

21 =3 \cdot 7

\therefore 777 \cdot 365 \cdot 21 = 7 \cdot 3 \cdot 37 \cdot 5 \cdot 73 \cdot 3 \cdot 7

Math is beautiful, isn’t it?

Remarks

The joke has made into wikipedia LOL

777.png

Fibonacci Number & Golden Ratio

Posted: March 22, 2017 in Mathematics

Today I saw this funny picture of Donald Trump and then I know why everybody says Golden Ratio can be found all over nature. Yes it is living among us. You can’t escape from it, not even from the president of America.

trump-golden-ratio-fibonacci.jpg

So, what is Golden Ratio?

In Euclid’s Elements, Book VI Definition 2, Euclid wrote (in Greek of course):

Ακρον καὶ μέσον λόγον εὐθεῖα τετμῆσθαι λέγεται, ὅταν ᾖ ὡς ἡ ὅλη πρὸς τὸ μεῖζον τμῆμα, οὕτως τὸ μεῖζον πρὸς τὸ ἔλαττὸν

Or in English,

A straight-line is said to have been cut in extreme and mean ratio when as the whole is to the greater segment so the greater (segment is) to the lesser.

Or in plain English, it basically says that when you divide a line segment into two uneven pieces, the ratio of the whole line segment to the longer piece equals the ratio of the longer piece to the shorter piece. And this ratio is called the Golden Ratio (often denote as \phi).

DivineProportion

\displaystyle\frac{a+b}{a} = \frac{a}{b} = \phi \quad\Leftrightarrow\quad 1+\frac{1}{\phi}=\phi \quad\Leftrightarrow\quad \phi^2=\phi+1

After solving the quadratic equation, you will get:

\displaystyle\phi = \frac{1+\sqrt5}{2} \approx 1.61803398874989\ldots

Now, let’s consider the Fibonacci sequence.

F_n = 1,1,2,3,5,8,13,21,34,55,89,\ldots

and examine some nth power of the golden ratio.

\phi^1 = \phi

\phi^2 = \phi+1

\phi^3 = \phi^2+\phi = 2\phi + 1

\phi^4 = 2\phi^2+\phi = 2(\phi+1) + \phi = 3\phi+2

\phi^5 = 3\phi^2+2\phi = 3(\phi+1) + 2\phi = 5\phi + 3

\phi^6 = 5\phi^2+3\phi = 5(\phi + 1) + 3\phi = 8\phi + 5

This suggests that

x^2=x+1 \quad \Rightarrow \quad x^n = F_n x + F_{n-1} \quad\forall n\in\mathbb{Z}^+

Let’s define F_0 = 0, then

x^1 = F_1 x + F_0 = x

x^2 = F_2 x + F_1 = x+1

Assume x^k = F_k x + F_{k-1}, then

x^{k+1} = x^k\cdot x = F_k x^2 + F_{k-1} x \\= F_k(x+1)+F_{k-1}x = (F_k+F_{k-1})x+F_k \\ = F_{k+1} x + F_k

Hence, by math induction, x^n = F_n x + F_{n-1} \quad \forall n\in\mathbb{Z}^+ \text{ where } x^2=x+1.

Now, go back to the equation x^2 = x+1, the two roots are

\displaystyle\phi = \frac{1+\sqrt5}{2} \quad\text{and}\quad \Phi=\frac{1-\sqrt5}{2}

Hence,

\phi^n = F_n\phi + F_{n-1} \qquad \text{(1)}

\Phi^n = F_n\Phi + F_{n-1} \qquad \text{(2)}

(1) – (2) yields

\displaystyle F_n = \frac{\phi^n - \Phi^n}{\phi - \Phi} = \frac{\phi^n - \Phi^n} {\sqrt5}

Or

\displaystyle F_n = \frac{1}{\sqrt5}\left(\left(\frac{1+\sqrt5}{2}\right)^n-\left(\frac{1-\sqrt5}{2}\right)^n\right)

Remarks

Since

\displaystyle\Phi = 1-\phi \quad\text{and}\quad \phi = 1+\frac{1}{\phi}

then

\displaystyle F_n = \frac{\phi^n - (1-\phi)^n}{\sqrt5} = \frac{\phi^n-(-\frac{1}{\phi})^n}{\sqrt5}

\displaystyle\lim_{n\to\infty}\frac{F_{n+1}}{F_n} = \lim_{n\to\infty} \frac{\phi^{n+1}-(-\frac{1}{\phi})^{n+1}}{\sqrt5} \cdot \frac{\sqrt5}{\phi^n-(-\frac{1}{\phi})^n}

As \displaystyle n\to\infty, \quad -\frac{1}{\phi^{n+1}} \text{ and } -\frac{1}{\phi^n} \to 0

\displaystyle \therefore \lim_{n\to\infty} \frac{F_{n+1}}{F_n} = \phi

As n\to\infty, \quad \displaystyle -\frac{1}{\phi^n} \to 0

\displaystyle \therefore F_n \approx \frac{\phi^n}{\sqrt5} as n is large.

The last approximation comes in handy when you want to find the nth Fibonacci number, let’s say the 30th Fibonacci number.

\displaystyle F_{30} \approx \frac{\phi^{30}}{\sqrt5} \approx 832040.00000024\ldots

So, you can compute the 30th Fibonacci number to be 832040 in seconds with any normal scientific calculator.

Einstein’s Five House Riddle

Posted: March 15, 2017 in Mathematics

This riddle is claimed to be written by Einstein when he was a boy. He claimed that only 2% of the people can solve it. Are you that 2% who can solve it?delightful-education-is-not-learning-of-the-facts-famous-quote-by-albert-einstein-training-the-mind-to-think-school-1024x768

There are five houses in five different colors in a row. In each house lives a person with a different nationality. The five owners drink a certain type of beverage, smoke a certain brand of cigar, and keep a certain pet. No owners have the same pet, smoke the same brand of cigar, or drink the same beverage.

The question is: Who owns the fish?

Hints:

  1. The Brit lives in the red house.
  2. The Swede keeps dogs as pets.
  3. The Dane drinks tea.
  4. The green house is on the left of the white house.
  5. The green house’s owner drinks coffee.
  6. The owner who smokes Pall Mall rears birds.
  7. The owner of the yellow house smokes Dunhill.
  8. The owner living in the center house drinks milk.
  9. The Norwegian lives in the first house.
  10. The owner who smokes blends lives next to the one who keeps cats.
  11. The owner who keeps horses lives next to the man who smokes Dunhill.
  12. The owner who smokes BlueMaster drinks beer.
  13. The German smokes Prince.
  14. The Norwegian lives next to the blue house.
  15. The man who smokes blend has a neighbor who drinks water.