七國民華中
育教 日八初月九年戊曆
華僑經濟
市股
烈激
資投
止中
附一之後
黑色星
应勰市堪忍
旺蠍烈的设
收縮的劈 來
W),就已終點盤戰火
是企圖把本身的產品有洲於其他規有輕細的經 一些模構投資者墡柏克萊出這樣一,
象-在菜躇 之後-棆敦股票市场的成交低至今仍未恢復治 檢視一下有關敷字便知道,去年十月股災
日企業倒閉
九月份再降
數字較去年同期嫁百分之二十四,五。
(*)
商研究所發
確:日本工
没九月份日
1989 中學會考預習專欄
*明德出版社
MILE & DALF PRESS
{k+1} {k+2}{k+ 3 } [ 2k +4 )
{* + ] [ {k+1} −1] [{k+1} +2 ]
[3ík+11+11
Pix+1) is also true..
By the principle of mathematical induction, Pla) is true for dll positive integers n
Additional Mathematics (3)
2.
Prove by mathematical
W. S. So
Revision Exercise 3: Mathematical
Induction
Worked Examples:
Prove by
that
mathematical induction
*sin(n+1}{n+2)(3n+1!
for all positive inteners n
Proof: Let Pin) be the statement
- 1 2 - 2 + 22 = 3+ + 2 (n+1)
== 2n{n+1}{\=2}{3n+1}
when n=1
1.1.5.=12.2=2
(R_H73) == {{( 1 ) (2) (3) (4)-2
Assure P(k) is true.
1.3. 12 2+..+k2(k+1)
*zzk (k+1} { k+ 2 ) ( $k+1}
when n=k+1
4...S.
- 1 2 - 2 - 2 2 - 3 + - - + k (k+1), +(k+1)2(k+2) *k+1}(x+2)(3k+1)
==2{k+1} {k+2}{k{3<+1)+12(x-1)].
{ k + 1 ) ( k + 2 } { 3 < 2 + 1 3 x +121
induction
that, 3-1 is divisible by 13 for all positive integers n.
Proof; when -1
3-1-26 is divisible by 13, it is true for nzi
Assume it is true for n-k
i.e. 3-1-13mm is an integer when nakit
=33.33k_i+1]-1- -333-13+27-1 =2/(13)m+26
43, is divisible
by 12
It is true for n=kaí
ay the principle of mathematicsi induction
33-1 is divisible by 13 for all positive integer n.
Revision Lxercise 1
Section A
1. Prove by mathematical induccion
that
2.
1.1.
... for all positive
√ √2
WR
integers ..
Prove that the sum of cubes of 3 consecutive positive integers is divisible by 9.
報日僑華
==n*(n+1}5 for ali_positive
integers n.
when n=2
4.
Prove by mathematical induction
· cósē[email protected]"-la-gina"e
. for ali positive. Integers "
Section B
5. Let, be the two roots of quadratic equation x2+px+p2=0, where p is a integer. (3) show that
is a multiple of g
(i)
+ is a multiple of p1
(b) (1) Find the value of
k+1.
£
(1) Herce show that
后
=0
(c) show by mathematical induction
that, for any positive integers 0.
Rip is a multile of pn
Solutions to Revision Exercise 3
1. When n=1, L.H.S.=1=T=R.H.S.
it is true for n=1
Assume it is true for n=k,
i.e.
啦
#her_a=k+1
2. H.S.=
你可
it is true for nok+i.
k+1 √k+1
By the principle M. I. it is true for all positive integers n.
3. Prove by mathematical induction that 2. Let Ping he
12 (143x^2)+23[+8x22)-
二期星日八十月十八八九一)年七十七國民華中 ( 16 )
13+23+33-25914}, {2} is true..
Assumie P(k) is true
ize. (k-1)343-(k+1) 39m
a is an integer
when_3=k+1
̇k2+(k+1)3+ (k+2} 3
***+{k+1}®+[1k+1)+319
=k3 + ( x + 1)2 + (k − 1 )2 +9 (k-1)2+27(x-1)+27
- 9m+9(k−1)2+9(3] {k~1) +9{3}
-9, is divisible by 9
.. P(k-1) is alse true.
By the principle of N. 1. P{n} is srue for all positive integers n>2.
3. When n=1
4.
L.H.S=1{1+3}=4;R.H.S% 2(1)(2)3:4
It is true for n=1.
Assume it is true for n=k
i.e, 13(1+3x12)+23(1+3×22)+
+k3 (1+3k2)=k3(k+1) 3
when n=k+1
L«H» S. = 1 3 { 1 + 3x : 2 } » . . . »k3(t=3k2;
(k+1)? (k+2) 3
Its true for m-k+1.
By the principle of M.I. it is true for all positive integers n.
When n=1
1.H.3.-cose
*sin29_2siu6cos8:059 R.H.S.25100 25119
it is true for a-1.
Assume it is true for rek.
1.e. [email protected]
2 sine
cosecos28cusce..
@cos2@
statement "(ng
Avisible by
aksine
5. He
Ming an
6. The job of a teacher is
{reward)
9.
Smoking
ile
to health.
of winning.
7.
7.
3
singing
On Sundays, buses ang trams are crowded. little)
10.
the inconvenient L
xt month.
8.
She fainted at the of blood, see)
11.
B. Your kind help will be
appreciated.
(great)
9. One should keep oneself and happy.
(health)...
2sine ecosz
2-2 sine
Xr'sinG
5.
(a) +8=-=2
sin8
It is true for n=x+1.
By M.l. it is true for all positive
integers n.
{1}]}=(+6) 7~263 ~p2 -25? --p" is
a multiple of p.
+=(x+} {*~£$ +5? }
_ - {~p} { -p2 <p2 }
#2p* is a multiple ps
[1]
k
(b) (i) x
-
since is a root of
·x2+px+p2=0.
=0
(11)from b11)3+1, 12 5 + 1 + k + ? )
$
-0+0
(c) when nal
2+6=-p, .'. it is true for a-1 from(at), (all) IL is crue for n=2,3
Assume it is true for nek-1 and nak.
from {b} ck+1+k+1
=-bp
<+1 +1
pk+1(mb-a).is a multiple of
k+1
it is true for ask-1.
By M.I. it is true for all
positive integer.
sidering the
ay timough A
incipal axis.
Cni [Ans.)
1989 中學會考預習專欄: 明达出版社
MILL & DALE PRESS
'English (3)
Dear Students.
Today's issue is an exercise on! parts of speech. Very often students
confuse. their usage and so it's worth having more practice.
(A) Write down the correct form of the
word given in brackets.
1. Children should be.
2.
their parents.
(obey)
The accident happened because the. driver's care)
3. The response from the audience
(courage).
4. They have not yet
(conclude)
5. Jane has taken
COME
门
midnight now. The children are
sleep!
10. The show was a great
(succeed).
34 For further, please contact Mr.
Wong during office hours
hurt
10. The gangsters demand noney from the
hawkers. (protect)
(C) In each sentence below, there is
one mistake. Underline the wrong and replace it with another that best completes the meaning.
The book is interested and I like it.
You ft hand in your work next Monday at the lates
12. Take 3 seat confort:
13. My boss is sak performance.
3. Further information), please
4. Fortunately), no
5. no (objection) to
5. but (rowarcing),
7. ora Gess! crowded
8. the (signtio
9. oneself (healthy) and 10. demand (protection) money
interested (interesting!
(saying).
2. said
(C)
14
Your advise has helped slut in solving the problem,
3.
15, 1 saw John Deat by a group of
gangsters.
Answer's
(A) 1. obedient
2. carelessness
3. ercouraging
4. conclusion
5. cockery
6. application
7. competition.
ARIES
objection (oo)jective)
4. Applications (Auplicants}
5. Füss.
6. cheated
7. died
8, hazard
9.
(Past)
(cheating)
{head)/are (have)
(hazardous}
confidence [confident).
10. inconvenient. (inconvenience) 11. late
12, confort
{latest}' '
「comfortable)
13. satisfactory (satisfied)
15. advise
15.
seat.
(advice)
(bauten)
(0) In each uf the sentence bakow, there is one word left out.. Write down the correct forw of the missing word by using the word given in brackets.
1.
2.
王 left the party without said goodbye.
1. We should put aside some money for
days. {rain)
.3.
You had start revising your book, Exam is coming. (good)
The objection of this competition is to arouse students' interest in writing,
1.4.
Applications should be male, aged between 30 and 40.
(inform)
5.
It is now half pass nine.
6. greatly:
one was
in
car
6.
The teacher caught the student
9. asleep
accident. (fortune)
cheated in the examination.
10.'succesS
[cook)
as one
no to your suggestion,
7.
of her subjects in FLA.
He is an orohan.. His parents are {B) 1.for(ran died.
2. had
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locating
ge of A (1.e.
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incipal axis}
f the lens can
in the graph).
cm (Ans.).
ual image
and the lens,
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