(20分)3條maths題…等差級數&近似值…

for question 1, 我不知道你想近似值去到幾多有效數字, 如果你學附加數, 微分學, 那麼
1. 1.004 ^3 = 1.004 x 3 = 1.012
1.004 ^2 ~= 1.004 x 2 = 1.008
0.7 x 1.012 = 0.7084

15000x(0.3+ 0.7x1.004(3次方) / (除) 1+1.004+1.004(2次方)
= 15000 x ( 0.3 + 0.7084 ) / ( 1 + 1.004 + 1.008 )

= 15000 x ( 1 ) / ( 3 )

= 5000
但如果不是附加數, 那麼你要用計數機, 得出數值, 並用三個有效數字.
計數機的答案是: 5022.053227, 三個有效數字(3 sig fig)即5020.

2. 利用公式: n 項總和 = (n/2)[2a+(n-1)d] (a是頭項 n是項數 d是等差)
we know that a = 25, d = 18-25 = -7.
so (n/2)[2(25)+(n-1)(-7)] = -830
7n^2 - 57n - 1660 = 0
(n-20)(7n + 83) = 0
n = 20 or n = -83/7(rej.)

3.(a) 3rd row = a + 2d
11th row = a + 10d

相差 = 第十一行-第三行
126-54 = (a+10d) - (a+2d)
8d = 72
d = 9
每行相差9個位

b) we know that a+2d = 54
a+2(9) = 54
a = 36
利用公式: n 項總和 = (n/2)[2a+(n-1)d]
共有座位 = (15/2)[2(36)+(15-1)(9)] = 1485

c) a+(n-1)d = 3a
(n-1)d=2a
(n-1)9=2(36)
n=9
第九行是第一行座位數目的3倍

d)增加座位=( a + (16-1)d ) + ( a + (17-1)d ) + ( a + (18-1)d )
=3a +48d
=3(36) + 48(9)
= 540

e)利用公式: (增加/原數)*100%
(540/1485)* 100% = 36.4% (3 sig fig) or 400/11%(exact value)

1. 1.004 ^3 ~= 1.004 x 3 ~= 1.012
similarly
1.004 ^2 ~= 1.004 x 2 ~= 1.008

0.7 x 1.012 = 0.7084i


15000x(0.3+ 0.7x1.004(3次方) / (除) 1+1.004+1.004(2次方)

==> 15000 x ( 0.3 + 0.7084 ) / ( 1 + 1.004 + 1.008 )

~= 15000 x ( 1 ) / ( 3 )

~= 5000


2. 25 + 18 + 11 + 4 + .... = -830

==> (25-7x0) + (25-7x1) + (25-7x2) + (25-7x3) + .... + (25-7x(n-1)) = -830

==> 25n + (-7)x(n)x(n-1)/2 = -830

==> 7(nxn) - 50n -7n -1660 = 0

==> ( 7n + 83 ) x ( n - 20 ) = 0

==> n = -83/7 or +20

i.e. total item is 20(terms)


3. Assume
1st row = a + ( 1 -1 )d
2nd row = a + ( 2 -1 )d
...
15th row = a + ( 15 - 1 )d

a) d (difference ) = 11th row - 3rd row
==> (a+(11-1)d) - (a+(3-1)d) = 126-54
8d = 72
d = 9

b) Total seat =
( 1st row + 15th row )/2 x 15
a = 126 - (11-1)(9)
= 36

( 36 + 36 + (14x9) ) /2 x 15
= 1485

c) (n-1)d+a = 3a
(n-1)d=2a
n-1=2a/d
n=2(36/9)+1
=9

d) ( a + (16-1)d ) + ( a + (17-1)d ) + ( a + (18-1)d )
3a +48d
==> 3x(36) + 48x(9)
==> 108+432 = 540

e) 540/1485 * 100% = 36.36%