1967-02-28 — Page 23

日十二月正年朱丁唐豆:

英文中學會考試題預習專權

學科 (十八)

·歐陽鎧文·

TLESSON

/MATHEMATICS (18)|

"PROGRESSIONS

"A' set of numbers, each of which is formed fṛóm one or more of the preceding according to some fixed laws, is called a series, The Successive quantities are called terms of the series, ARITHMETICAL PROGRESSION :(A.P.J

A series is called an Arithmetical Progression{abbreviation? A.P.) if each of its term is formed from the preceding by adding ho it a constant quantity (may be positive of negative). And this constant quantity is called the common difference. STANDARD FORN: a, atd, a+2d, ... 、、*

in which, the nth term-1=a+ (0-1)d.

the sum of n terma-S=(a+l}

EXAMPLE 1: The 3rd and 6th terms of an 8.P. are 28 3 37 resp.)

find the series.

SOLUTION: With the usual notatión,

The 3rd term = a + 24 = 28

The 5th terne'a + 5d=37

nence by subtraction, 3d=9, Xam),

r Subat, d=3 into the first equation, a +(2)(3)=28, -Tam 22 19

the required sories is 22. 25, 28, 32.

NOTE: An A.P.ls completely determined when any two terms are?

giveny

ARITHMETIC WEANS:

(2) Then three, mumbers are in A.P., the middle term. Is" called

bha arithmetic mean of the other two.

(2) When any number of numbers are in A.P..the terms Inter-,

mediate between the first and the last are called the

· arithmetic means between these two terms.

Hence, it is always possible to insert any required number of arithmetic meanë between two given numbers.

EXAMPIE 2-Theert n arithmetic means between x &y.

LUTION; Including the given terms &, there will de (n=2)

terms in «P. of which x is the first and is the last. Let as the common difference.

Then, the (n+2}}} tern=x+[(n+2}-y a = 9

the required. means are:

(***

kala, 24 Apri

n+1

EXAMPLE 1:If p, 5p, 6p+9 ars in a.P., find : and continue tha

series for 4 terms,

SOLUTION

P, Sp, op79 are in A.P.

the common difference (6p+9)-5pm 5p-p

the common difference = 4p=12

the series is

P+ 94D

3412, 3+122, 3123, 3+1284, 3+1245, 34 1288,

The following notation fe, sometimes convenient:

1) ine successive terms may be denoted by

mere the soffix indicates the number of the term in the seria The sum of any assigned number of terms may be denoted by S in a suitable suffix number dig. 5., represents äum to 20.

while stands for "sun to a terme a

Lt Pind the gun of (a) the first n integera, (b) thế, firat n even integers, (t) the first n odd integers

For the set of the first n integera

the 1st term = 1

the laat term

no, or

torns en

"the aumeiw

(b)for the set of the first nev

the 1st terg=237

the common difference = 2

the number of terms-n

the eumddn

c)For the set of the first n odd integers

the lattern = 1

the common differences

the number of teras – n..

the aume3e7f 2m1 4(n-2)*2] = a*

MONG

EXAMPLE 5: The sum of 5 numbere in A.P. 18.30, and the sum đề their squares is 220; find the numbers.

SOLUTION: Let a be the middle no, &'d be the common difference.

Tuan, the £ hos. are! a-20, &-0

Henge, their summ5as30

The sq, of the no.are: (-2d

Their sum-2(a* + 4d*

WAH KILLY CITY HALL

reciprocale of its terms are in!! PExamples 16 HP,

usually solved by inverting the terms and using the proper, ties of its corresponding A.P. There la fio Eneral formula for the sum of a number of terms in H þ

GEOMETRIC PROGRESSION (G.P.)

A series in which each term is formed from the preceding bi multiplying it by a constant factor is called a Geometrit Progression, Abbrev. G.P.) The constant factor 10 mure

often called the common ratio and is found by dividing ang term by the tarz which precedes it,/

STANDARD FORM: 8, at, ar3,

in which the nth tere La är

GEOMETRIC MEANS:

(1)hah: thrss "numbere?growan. C.Pithe dddle to la called

the geometric mean of the other two.

(2) When any number of numbers are in C. P. the terms inter-l

mediate between the first and the last are called the geometric means between the two givan terms,

EXAMPLE 8. Find the geometric mean 6ft x & gu

SQLUTION: LL G be the required mean; then,xiGoy, are in d.P

Hence, the common ratio =

C2m xy, t

Gmi

The geometric dean of x & y 18 therefore /zy.

NOTE: It is usual to take positive aquare foot.

NOTE: IF À ̧GH, are the arithmetic,

between two givan terne X &

ALEX

AndAH Lith

geometric,and harmonic means)

have proved that

1.6. & is also itself the geometric mean between A an

EXAMPLE 9: Find two numbers such that their arithmetic meal i

25, and their geometric mean 18 24,

SOLUTION: Let x &y be the

Then, the arithmetic mer

the geometric, mean (x-y) = 196

{1} - ( 2 ):

Hence)x=32, ye 18.0

gumbers.

ANS. THG wo numbers Ar32.18

11

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日八十月一年七六九一股公年六十五國路

繫。雷健行助本有新天

所約 領羊 校

,蒼香新如施

姜奉授教昻李

50%

任主理物大港

·576

(3)

'EXAMPLE 10: The 4th, Bth, and 24th foren of on A‚E, are in 0.7.)

find the common ratio of the OP

SOLUTION: Let the 1st tera, o da the common difference;

Then, the ith térn as 30

the 6th term EB470–

the 24th tera ea +240)

447d, and a +3d are in dif

4+78 1# (+238)

funless d0-

the commish.ratio in d.f

Find the aim to ʼn terma,

SOLLETON in the 0.P.

· of terms = fà,

Thus, it appears that however many terma we take, the sum ja alwage less thần 2. and, by taking n sufficiently large, we can fake the fraction

68 amall as we please. In other words.

by such n the sum car amall as we please.

be made to differ from 2 by a quantity as

"Hence, in the standard sefiða – é • arvát

EXAMPLE12: Express 7 SOLUTION

2857,28

sé a common fraction: 85.85

in the G.P.

EXAMPLE 13: How many terma of the G.P. 0.8,

1.2, amat be taken to give a aum greater than 16602 SCLUTION che aim of ntersus = 08x4504 Now w ind the smallest possible value for Insoumlity

|.5*~ 1 > 1600 = (-1000)

10g 1.5 16g1001

IBX

satisfy the

港抵目昨於人夫與經

兴湖方物

·李被投在尼日尼维伊岛丹大学校 李昻敎 颺於愛丁大班,一 。李希激授偕夫人已於昨日抵港任職 *一九六三年任蘭經學灏授,年任 九五六年取得博士學位。一九五五年

港九新區天台校聯會籌辦

員生保健計劃

正與宗教福利醫療機構恰活中

1,直背保曼,均能把"教育委員決月||中學及小學會考人數

逐年作直綫上升

(上)有纪兔盲文上午小图高年級家長教師座機會,個爲校長王饰模致詞。而下,有安M官立小学高年 *有舞本 英文單一到粉考試委員會已,後者能在難之英將免被美國大季能爲外有等大學亦持此酯 出算幾計分。英九六五年民始,英國大學之人頭試的「良一科及以上緞,超越入學試及格,與

數

級家長敎師座談會,圆爲家長珽說師座談涔槽形。

「國情况方面:

八時半誥,隨

一的意見。該區

中

會

障

分

美國機

真在使詔英:

多成此

雞實中

新烈天合理錢著的,盡,將來實施後,凡 過疾病,醫吳毅之無「包高榮,然榮帶花道 《中國 九 南學生反被師保健計校教師待遇微薄,如 x xx之家長」之江 下,於門道口一般家長查网偶做股神舉行茶會招待,九個,因此另類國之

來

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j爲一萬一千三百人,因此,一九空

,家長逐個扣上區

該會所擬推行一系到之稱。良以私案眾。

家長教師座談會

談資以阿學位,分 、李玉藏等。查被座 行,铈 談

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本身我從奉委爲骸大学物钸系主任,港大校外部中午講座 (特杌 香蓓大學消息,猜塞,千人。参加老國人獄·文中學會考,將用

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(原团,苍港大學校外蹀區高主謀中

由聯合書院高級斄師掞 主講 歡迎各界蒞

个人公開前座,选星期三中午一時十五分,

唐詩欣賞

the required numbeds are 24,6810,

NOTE: When we have to find an even number of terms

it, is best.

to take and, and for the two midfie terma, në that 2d 18 the common difference,

Two men set out te mest each other from two places 165 al. apart, One Gravels 15 miles the firat day, 14 the second, 13 the third, and so on. The other travela 10 miles the first day, 12 the second, 14 the third, and so on When will they meet? SOLUTION: Suppose they will ceat each other in n days after,

they start.

For the 1st man, he travela: 15m1..14m), 13md

which corresponds to the A.F 15, 14, 13. in which, a 15,6-1, no. of terms

f

.*7[2a + (n-1)d] => [ 2015 + (n-1)(-1)] = 2{31-6). for the 2nd man, he travels 10m1, 12m1 1481

which is equivalent to the A.P. 10, 12, 14.1. an which am 10, do 2, „ no. of termsony

9 ̧¤^{2x10 + (n−1}{2}} ~ n[10 + (n=1)] sTb[9 +

ANS

(31−n) + n(9+ n) = 165. (Total distance.

n(31-n) + 2n(9+)= 330

A* +498. - 3300

(+55)(n = 6)~0

•n-6 or

b (rejacked)

that)

each other are in A.P. and b,øjd in h,

17b4 0) = 268

2bd-ad = 2bd

(a + 8 } bì að + ad

P. is the abbreviation for Harmonie Progrèssion. series in said to be in Harmonic Progressiób, if the

the smalleat, possible value of n 15 18.

17.03.

is came aust be. taken to give a dm greater than 1600.

NOTE: Many questions involving G.P. (eig, comboŭnd Interest,

annuities, Pepayments by squal instalmentă, insurances ett, are best solved with the aid of logarithms.

HINTS & ANS, TO EX”-17

Solution

Suppede We man can cycle straight forward to the station than han st. line (a) Suppose man can catch the train on foot only on time, we find st Ime cbs.

uns The

chan of lines) (a) and (b) is the pot. tr.

Solution [When yea, x=

Max.

value of

x26F72314 5167 EX 760 KG 1818 24 30

ZUM 4.9 16 25 36 48

·TANS

starting-point

They were

miles apart

.16

4

EXERCISE 18

Find the arithmetic weari of reg

2) Find the series in which the 5th & 21st teras are 7K-By

23x-40y.

31 The term of the series 36+ 20, 56 +6, 76;

176

-

Se; find n.

4) How many terms of the series 42, 34, jb,

that the Bum máy be 312 *

5) Find the sum of x arithmetic means betwsan x and-)x. Fine also

the common difference.

6) Insert ʼn geometric means Between x 73.

7) Flad the sum of

(0) A man puta by for his son on every 617thday a half-erom for

every year of his age, How old will the son be when the total

utput by amounts to £17.2

9) The middle points of the sides of an equilateral triangle are

Joined, formong a ascend

a third triangle la formad

by joining the middle points of the second, and the process is continued indefinitely. If the perimeter and area of the orig- inad triangle ars på A raspy; find (1), the sum of the paris. meters, (il) the sum of the areas. of all the triangles,