1969-03-04 — Page 14

算二第張四第日六十月正年西已靥夏 WAH KIU YAT PO

郭日僑華 報日橋

二期星 日四月三年九六九一圈公年八十五國民中

★教僑華

[CC文中學會考試題預習

數學科

·歐陽 女

MATHEUTICS (18).

LESSON 18: PROCRESSIONS

A set of numbers, each of which is formed from anes or more of the preceding according to some fixed law," is called a series. The successive quantities are called terms of the series

rithmetical Progression:( A. P.).

A series is called an arithmetical Progression. Abbreviation: A.P.) if each of its term is formed from the preceding by adding to it a constant quantity may be ve orve). And this constant quantity is alled the common difference.

STANDARD Form:20, 2424,

In which, the nth terma+(n−1)d ̧ ́

and the sum of n te

=-L[2a+(n-1)d]

EXAMPLE 1: The 4th and 8th terms of an A.Pare 31

and 43 respectively find the series.

SOLUTION: With the usual, notation.

the 4th term ar4d-31

the Sth term a48d43

Whence, by subtraction, 4d=12. Subst. d-3 into the 1st equation::

4(3)=371

22

{Thus, the req'd series is: 22, 25, 28

NOTE: An A.F. is completely determined when any two

terms are given.

Arithmetic Means:

(1) When three nos, are in A.P., the middle term 18.

called the arithmetic mean of the other two.

(2) When any no. of nos, are in A.P., the terms inter-

mediate between the first and the last are called the arithmetic means-between these two terms. (3) It is always possible to insert any required no.

arithmetic means between two given nos...

EXAMPLE 2- Insert n arithmetic means between x'otul SOL : Including the given terns x and y, there will be

(n+2) terms in an A.P. of which x is the first and y is the last

Let d the common azference

Then, the (n+2)th_term=x+[(n+2}-1}đ=— y

NOTE: When we have to find an even no. of terms,

is best to take a-d, aid for the two middle terms, so that 2d is the common difference

EXAMPLE 6: Two men set out to meet each other from

two places 165mi, apart. One travels 15 miles the first day, 14 the second, 13 the third, and so on. The other travels 10 miles the first day, 12 the second 14 the third, and 50 on. Wher will they meet?

Suppose they will meet each other in n days after they started

For the first man, he travels 15 má, 14m1. 13 mi..

Which corresponds to the A.P. 15, 14; 13, In which a 15, de-1, no. of terms-n

= £[20+(n=1#F

= 4[2x15+ (n-oc-ol

(31-n)

For the 2nd man, he travels 10mi,12 my, Which is equivalent to the A.P. 10, 12, 14, In which

a-10. d2 no. of terms

- 4 [2x10+ (n-1}{2})

n(9+n)

unless a

the common ratio in G.P.

Id

EXAMPLE 11: Find the sun

G.R.

to n terms

of the

SOL

In the G.P. a-1, It, no. of ternis n

رگ

Thus, it appears that, however many terms we take, the sun is always less than 2. And, by taking n. sufficiently large, we can make the fraction

as small as we please. In other words, by such n, the sun can be made to differ from 2 by a quantity as small as we please.

Hence, in the standard series, atarta

ar

Sn

a(1-x)

1-r

-6 or #55 (rejected)

EXAMPLE 7: If a, b, c are in AP. and b, c,

prove that ad bc

NOW, 13 Te☺, rù as niner

The expression -r is called the limit of the sum (or the sum to infinity), and is usually denoted by

S=

EXAMPLE 12: Express 7.285 as a common fraction.

SOL.

7.2857.285 85 85. J

7+

85

10 1000 100000 10000000

Proof:

£ (3)•n}+n(9+h) n(31-a)+2n(9+) 330

165(Total dist.

*+49n-330 0 (n+55)(d-6)=0

They will meet each other in 6 days

b, care in A.P.

d are in HP.

subst

2b = a+ 40S

•{2b-a)(b+d) 2bd 2b*-ab+2bd-ad- 2bd

But 2b-a-c

2babrad

(asc)b abrad ababe — abrad bc on ad

2b

b+d

Note: H.P. is the abbreviation for Harmonic Progression.

A series is said to be in H.P., when their reciprocals are in A.P. Examples in H¿Pare.

usually solved by inverting the terms and using the properties of the corresninding A.P. There is no general formula for the sum of a no. of terms, in H.P

GEOMETRIC PROGRESSION: (G.P.)

3 series in which each term is formed from the preceding by multiplying it by a constant factor is called a Geometic Progression (Abbr. G.P.) The constant factor is more often called th common ratio, and is found by dividing any by the term which preceeds it

standard form:

a ar ara In whichl, the nth term

mark=!

the sum of n terms S.

the G. P.

7.285

-7+16+1000 (+ 100 *10000

7+ 10 + 10x ( 1 + 10* + 10**

99

940

287

EXAMPLE 13: How many terms of the G.P. 0.8.1.2, 1.

Tu must be taken to give a sum grea than 1600 ?

For the G.P. 0.8, 1.2

The sum to n terms -

(1.5-1)

Now, we find the smallest possible value for

to satisfy the inequality

(1.5°-1) ≈ 1600

1.5"-1 > 1600 x-7(1.e. 1000).

Taking 105-

1.5" log 1.5

1001

log 1001

n>

10g 1001

Jog 1.5

2.0005 0.1761

The required means areï

• x+2(x-x),

LEXAMPLE 3: IT p, 5p, 6p+9 are in A.P. find p, and

continue the series for 4 terms.

** P,5p, 6p19 are in A.P.

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

_or_0+9 = 40 ***

the common difference = 4p =

the series ist.

3, 3+12, 3+12x2, 3+12x3, 3+12x4, 10. 3, 15, 27, 39, 51, 63, 75,

The following notations are sometimes convenient (1) The successive terms may be denoted by:

Tr

Where the suffix indicates the no. of the ter

in series.

(15) The sum of any assigned no. of terms may be

demoted by S with a suitable suffix no. *.g. Die represents "Sum to 16 terms", while S stands for *Sum to n terms".

EXAMPLE 4 Find the Sunsof (a) the first-n odd integers,

(b) the first n even integers, (c) the first

n integers.

SOL ) (a) For the first n odd integers,

the first term is 1

the common difference is 12

the number of terms is n

the sum≈ 5.

~~[2x1+(n-1)x2]

for the first n even integers

the first term is 2.

the common difference is the number of terms is

the sum➡S. —

Geometric Means:

(1) When three nos. are in G.F., the middle term

is called the geometric mean of the other two. (2) When any no. of nos, are in G.P., the terms intermediate between the first and the last are called the geometric means between these two even tarms.

EXAMPLE 8: Find the geometric mean of x and

SOL. Let G be the required mean

Then. x. G, y are in G.P.

Hence, the common ratio -

*xy for

The geometric mean of x and y is therefore √xy

(It is usual to take the +'ve square root)..

NOTE: 17 A, G, H are the arithmetic, geometric, and harmonic means between two given terms x & 3% we have proved that

A=

Arıd · All am (*~*~*^ ) ( − 2 =

n> 17.03.

the smallest possible value of n is 18

i.e. 18 term must be taken to give a sum >1600 NOTE: Many questions involving G.P. (e.g., the problems on compound interest etc.) are best solved with the aid of logarithm.

1.8.1

HINTS

Both functions are linear. You will find wwD STI

aphs, farm finct two tables of

line

DIST, (MI).

common point

-+(38-32)

For the first n integers,

the first term is 1

the last term is n

the no. of terms is'

the sum S ≈ (n+1)

EXAMPLE 5: The sum of 5 nos. in A.F. Ls 30, and the

sum of their -squares 16 220; find the nos.

(SOL Let a be the middle no. (of the 5 nos. in A.P.)

and let d be the common difference. Then, the 5 nos. are a-2d, a-d, a, a+d, a+2d.. Hence, their sum – 5a - 30

a

Sq. of the nos, are : (a-2d), (a-d)", a", (a+d)

4-2d)*

(Their sum = 2(a*+4d*}+2(a*+d®)+a”,

-5+100 = 220

the required nos

SOE.:

1.0. G is also, itself, the geometric mean

etween A and H

tix Lef

(EXAUPIE 9. Find two nos. such that their arithmetic

mean is 25, and their geometic mean is 24 Jet x, y be the two nos; Then, the arithmetic mean (x+y)=25) 1.6, x+y=50. (their geometric mean-xy 24

pile. x7 = 576-

(1)′′-{2}xb_{(x−y)”— 1961

Form (1) and (3), we find x=32,y=18)

ANS. The two nos. are 32, 16)

EXAMPLE 10: The 4th, 8th and 24tn terms of an A.P.

are in G-P., find the colamon ratio of the G.P.

as the lat term and d be the c. d. Then, the 4th term

SOL. Tet

18th term

a43d

aaïd

.24th term = a+238-

a+3d, as7d and atzia are in G.P.

(a+7d) == (a+3d)(a+23d),

Gd(3a65a).

a*+26ad+69d❤

posi!

the travel

"TIME" (min.)

ide with origin the first man

Girs At point (which is a point on X-axis and 8 min. apart from the origin), draw a line parallel to the travel graph of the fined man

(r) Then the point of intersection of

and the second man's travel

measuremunt, B 4 36

The graph of yo

a straight line

6x+4

Curve. USTALL

fline

x2= ±(6x+4)

are the point

taneous equation

The roots of 5X* = 6x+4

ordinates) of the sin

5x=6x+ta

Since

walled

y=$(6x+8) and

then y

ANS

are

(未完轉入第四張第