****
EXAMPLE 10: The at 8th`and"24th" tems" of an AP. We in G. A find we common ratio of the
GP
黃二第張四第
日九廿月正年申戊曆
WAH KIU YAT PO
報
白
二期
日七廿月二年八六九一公年七十五國民中
英文中學會考試題預習專欄|
4 420
A = 6
數學科
SOL
This
(†Ʌ)
歐陽錢文
MATHEMATICS (18)
PROGRESSIONS
So
NOTE When we have to find an even no of terms
best to take a-d. ard for the two middle, terms
that za is the common difference
16 15
འ
a.+ 3d,
ket, a m the 190 termļand -d be tha«C_4]
A
#e&# term = a+s4 8th term a+74
24th term =
a + 23 a
atid and at 23d are "GP)
(0 +7.2). = (Q+3α){@+234)
Q** 14ad +49α*= a*+ 26 ad $69d2
4A (3a +$4)= ←
2
sid
2474
A+34
→
otay 12 the second, When will they meet
in the third, and so on.;
Me commen Matto na GR=
음
A
халы
15
13 mat.
13
femmes = "
EXAMPLE II find the Sum,
← A SOL In the GP..
.. SA
1.
One
LESSON 18
A set of numbers, each of which is formed from
mare of the preceding according to some
JK
fixed fay, quantities are
called a series. The succesSÅVEL called terms of the series.
Arithmetical Progression (AP)
A semes is
PE33/00
called an arithmetical Progi
( Abbreviation (A P) if each of its tētu
is formed from the preceding by adding to it a constant quantity is may be this constant quantity is
difference STANDARD FORM
A,
a+d
In which the nth term = l
and the sum of n terms = $
EXAMBLE 18 The
"}
t're of 've). And called He common
Arad, S
S
Q+ (n-1)) d
7th and 8th terms of an A. P. are and 43 respectively, find the sexes
usual notation,
EXAMPLE 6 Two men set out to meet each other from two places 165 mi apart One travels 15 miles the first day, 14 the second, 13 the third, and
The other travels to mockes the first
so on
Sot. Suppose they will meet each other i
after they started
For the first man, he havels
which corresponds to the AP.
La
which
$, = 2 [+
4(2×18
棄しまい
For the 2nd man, he travels
which is
in which
squivalent to the Ai
a = 15, d = -
Q = 10.
#C942}
5,9
#(3-2)+n19+x3a165 (Total dist)
n{3}->]+ zn{q+n}= 330 ****? n. 330=0
GJ (+55}(-
# = 6 -55+ rej
Bands
They will meet Pack otheis in 6 days
EXAMPLE
Preet
If a, b,
H
The in A.P
bc
b. c
Q
15
d the in HP.
abd
eb xa+
By subst
S70, 1
reb-a){ brα) =bd ab-ab+ 2bd-od= Abo
• a b'x abtad Rut ab=# * c* ·· (a+c)bæ ab+ ad
et in
+7
unless deo.
to a ferms, of the
= 1. no. of terms, um 17.
* 2() ·
2-
we
can make
Thus, it appears that, however many terms jake, the sum is always less than 2. And, by taking a sufficiently large, we the fraction to as Small as we please.. In other words, by such n, the sum
be made to differ from 2 by
as Smal
we please.
ALL
Đ
can
guantity
Hence in the standard series, atare arta, tas"
MS
Now, if <} consequently
called
1-7
n pas
ت حس
arn
Increases ;
erd
The expression the limit of the sum
for the and is usually denoted by
Sen
Sand
as
to infinity),
Examris 12: Express 7.285 as SOL
common fraction
7 285 = 7.2 85 85 85 ......>
85 +
35 + (1+
+
落水
85
In the G.
a = 1
7.285 =
7+
85
100 OD
*
>
-
1.)
SOLUTION
With the
the
4th term
the
8th term} ma
a+44 31 a+&d=43
*=3
a = 72
NOTE
whence, by subtraction, 4d ≈ 12
stequations
Subst d=3 into the 15$
4+ 4(3) = 3/
Thus the rega semės is: 22, 85, 28, 31-
An AP is completely determined when any
am given
two terms
Arithmetic Means
+21
<> when three nos. are in A.P.. me mikate term called the arithmetic mean of the other two. When any no. of nos. are in AP., the terms inter. mediate between the first and the last are called the arithmetic means between these two terms It is always possible to insert aux repaired of arithmetic means between two given nos EXAMPLE 2. Insert n arithmetic means between » Sot Including the given terms x and y, there' A.P. of which a y is the last
difference
་
(n.) terms in on
first and ✰ the
be
the
Ket
Then the
common
th ferm
The required means are i
+
EXAMPLE 3. If 7 5p. 60+4 are in A.P..
Sel
and continue the sexes for 4 terms
$ 54. Cp+9 are in A P
*
common difference =(64+4)-53 ק
P = 3
.. לים
common difference — 4psil
The
Wha
series is ;
3.
3+1.
3+12x2, 3+1483, Smeat,
3 15. 27, 39, 1, 63, 75,
The following notations are sometimes convençkot s cia The successive terms may be denoted by:
7
Ta T
T-
Ta..
Tni
ab+bc = abrad
- be=ad
NOTE: H.P is the abbreviation for Harmonic Progression."
A series is said to be in H P., when their
reciprocals am M A.P
usually solved by 1 using the properties
There is no general
no. of terms in H B
males in H.P. are
•The haring ord
thesponding A P
fay the exism of a
GEOMETRIC PROGRESSION (GP)
A series in which each term is formed from
constant the preceding by multiplying it by a
Geometric Progression
factor is called o
1. Abby GP) The constant factor is
and often called the Common Natio
more
#S
found by dividing any ferm by the term which
preceeds if
Standard form
In which
A
art
a
The
the sum of n tems
ig at deren za ďa
w
or
KONG PUBLIC BR
where the suffix indicates the no of the tems in term dir The
Sum of any assigned no. of terms may in denotes by S
suitable suffix no e.g. Sie represen 'Sum to 16 terms', while S. stands for "kum to 'n termi,
VOITE
#
EXAMPLE 9. Find the sum of toy the first in odk
integers, (b) the fist
SOL
" even integers.
les the first
n integers (a) for the first " odd integer.
the first term is y the comnmn difference es number of terms u
Swar = S,
the
de the first a the first tem
the
2
even integers
common difference number of ferm. the sum - S; =
10 Že the fret
Z
喏
ኣ
12
»
intages »
73
+
+
the first term
+
the
the
tast · ferm no. of terms the sum = 5,≈
[
2
EXAMMA 5 › The, sum of 5 nos in R
器
30,
Sum of their squares is 220; find the nos
SAL
Let a BAR tot
oc
Šam Mary
in 41).
difference
be the middle no (of the 5 not
be the Then, the E nas. are a-2d, ad Hence, their Sum » 5a «jo op of the nas am: ca-sast, ca-ds" sa", cood), (art) - Their sum == 3{d°£44*)$ à [Q*$d*) + @
Geometric Mergs
When three mds
the
MmAdla
term is called the geometic mean of the other two 12) When
Ony terms
ho. of nos
are in Ĝ P, the intermediate between the tixe and the last we called the geometric means between these two given terms EXAMPLE & Find the geometric mean of x and y
SOL
Let G Then Nench
x
the
be the required mean
M 6. y
Common ratio G = Xy
G
меки
=
#
兽
地
ar
the geometic mean of x and y is then for NT I
is usual to take the
€24
I've squam reot).
NOTE: If A, G, H are the arithmetic, geometric, and
harmonic
means between two given terms
y, we have proved that
A 444, G-√%¥
And A# =(***)(***) = x * × Q^.
2+ y
G is also, itself the geönetic mean belwan
18 A and H.
EXAMPLE 3: Find two nos. such that their arith-
metic
Sol.
mean is 25, and their geometri
44.
mean is
ket x, y be the two nos Then, the arithmetic, mean =LU+y)» 25
*. 2+3=60 ------ (13) their geometric mean=√Xÿ=24
12. 13m876 -· **- . (a)
[#3′′ = €*>X4:
From (1) and cas, we find x=32, y=18 Ans. "The two nos ar Je 18
*
+
+
ཆ x
103
199 99
= 7+ 3+ 45
7-
EXAMPLE 13: How many ferms of the 9.1. 68, 12
must tag taken to give a Sum-
Der the G.P.
0.8, 1.2 1.8-...
greater than 1600 ?
The sum to n terms
1.5" - 1
-+
Sn
*** 0.8 X
0.5
(1.5"-17
Now, we hnd the smallest possible value for
to satisfy the inequality:
"$(1.5" - 1) > 1600
n
1.5" - 1 > 1600 x3 (14_Tova)
1.5" 100!
Taking tapi,
#
7 loy 15 » loy look
Log 1.5
3.0005
77.0
Mc
smallest possible value of ʼn is 18 18 terms must be taken to give a sum > 1600 Nets Many questions involving sm
DA
Compound itsast ak's are with the
and of logarithm
HINTS & ANs To Ex. 17
1. The distance between X
ond
Y is abbitvary
the Draw, by st line i
graphs for A's walk then for då cycling una ci
motoring
the lines o
a4 P as
both
concurre
shown
Y
+
(+9
best
the proto j solved
K
TIME Chr
8 and C pass A at the same place.
THO
2;
30
BLAU
(i) Let the position of A concide with the origin o
( Draw the havel graph of the first mon
Gia
"
Second
1- axis and $
line parallel
first man
At point P ( which is a point on min. apart from the origin), dram a to the trave! graph of the Then, the pt of intersection of this dot - line and the second man's travel graph is the positiu
of B
1
hom the graph we fine that B' is 36 me from n
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