國二數學1-2等差級數

首項 = a1
公差 = d
項數 = n
第 n 項 = an
首 n 項的和 = Sn


= = = = =
1.
a1 = 1
d = 3 - 1 = 2

an = a1 + (n - 1)d
29 = 1 + (n - 1)2
n = 15

Sn = n(a1 + an)/2
= 15(1 + 29)/2
= 225


= = = = =
2.
d = (1又1/3) - (2又1/6) = -5/6
a1 = 2又1/6 = 13/6

an = a1 + (n - 1)d
-2 = (2又1/6) + (n - 1)(-5/6)
(13/6) -(5/6)(n - 1) = -2
-(5/6)(n - 1) = -25/6
n - 1 = 5
n = 6

Sn = n(a1 + an)/2
= 6[(13/6) + (-2)]/2
= 1/2


= = = = =
3.
a1 = -105
d = (-95) - (-105) = 10

an = a1 + (n - 1)d
105 = -105 + (n - 1)10
10(n - 1) = 210
n - 1 = 21
n = 22

Sn = n(a1 + an)/2
= 22(-105 + 105)/2
= 0


= = = = =
4.
a1 = 3

Sn = n(a1 + a2)/2
n(3 + 83)/2 = 903
86n = 1806
n = 21

an = a1 + (n - 1)d
83 = 3 + (21 - 1)d
20d = 80
d = 4

項數 = 21, 公差 = 4


= = = = =
5.
a1 = 1
d = 4 - 1 = 3

Sn = n[2a1 + (n - 1)d]/2
92 = n[2(1) + (n - 1)3]/2
n(2 + 3n - 3) = 184
3n² - n + 184 = 0
(3n + 23)(n - 8) = 0
n = -23/3 (不合、捨棄) 或 n = 8

1.求等差級數1+3+5+7+......+29的和。
(1+29)15/2
=450/2
=225
A:225

2.求等差級數2又6分之1+1又3分之1+.......+(-2)=?
1又3分之1-2又6分之一=-6分之5
-2-2又6分之1=-6分之25
-6分之25除以-6分之5=5
5+1=6
[2又6分之1+(-2)]6/2
=1/2
=2分之1
A:2分之1

3.求等差級數(-105)+(-95)+(-85)+......+85+95+105=?
(105+105)/10+1=22
(-105+105)22/2=0
A:0

4.設一等差級數首項為3，末項為83，何薇903，求此等差級數的項數及公差。
設項數為x
(3+83)x/2=903
86x=1806
x=21
(83-3)/20=4
A:項數=21 公差=4

5.設等差級數1+4+7+10.......到n項的和92，求n的值。
[(n-1)3+1+1]n/2=92
(3n-3+2)n=184
(3n-1)n=184
3n²-1n-184=0
3n 23
n -8
(3n+23)(n-8)=0
n=8
A:8