數學一問(10分)

1.)在一等差數列中,某三個連續項的和為30,及首項與第三項的比例為3:7。求此三項的值。
設該三項為 (x - d) , x , (x + d) ,
x - d + x + x + d = 30
3x = 30
x = 10 ----- (1)
(x - d)/(x + d) = 3/7
7(x - d) = 3(x + d)
7x - 7d = 3x + 3d
4x = 10d ------ (2)
(1) - (2) ,
4(10) = 10d
d = 4
∴第一項是(10 - 4 =6) , 第二項是 (10) , 第三項是 (10 + 4 = 14)

2.在一等差數列中,某三個連續項的和為24及此三項的積為312。求兩組代表此三項的值。
設該三項為 (x - d) , x , (x + d) ,
x - d + x + x + d = 24
3x = 24
x = 8 ----- (1)
x(x - d)(x + d) = 312
x(x^2-d^2) = 312
x^3 - xd^2 = 312 ----- (2)
(1) - (2),
8^3 - 8d^2 = 312
512 - 312 = 8d^2
d^2 = 200/8 = 25
d = ±5
當d = 5, 該三項為 (8 - 5 = 3) , 8 , (8 + 5 = 13) 　→　3 , 8 , 13
當d = -5, 該三項為 (8 - (-5) = 13) , 8 , (8 + (-5) = 3)　→　13 , 8 , 3

第一個數 = X ; 差數 = Y ; 第二個數 = X + Y ; 第三個數 = X + 2Y
X+(X+Y)+(X+2Y) = 30
3X + 3Y = 30
3(X+Y) = 30
3(X+Y)/3 = 30/3
X + Y = 10
X = 10 - Y

X / (X+2Y) = 3 / 7
7X = 3 (X+2Y)
7X = 3X + 6Y
7X - 3X = 3X + 6Y - 3X
4X = 6Y
4(10 - Y) = 6Y
40 - 4Y = 6Y
6Y + 4Y = 40
10Y = 40
10Y / 10 = 40 / 10
Y = 4

X = 10 - Y
X = 10 - 4
X = 6

第一個數 = 6 ; 差數 = 4 ; 第二個數 = X + Y = 10 ; 第三個數 = X + 2Y = 14