diketahui a, a+b, dan a+5b merupakan 3 suku pertama suatu barisan geometri. jika a, a+b, x, y, z merupakan 5 suku pertama suatu berisan arimatika dan x+y+z = -1

Dik:
x + y + z = -15
a, a + b, a + 5b = 3 suku pertama
a, a + b, x, y, z = 5 suku pertama
Maka, x = a + 5b

Dit: U(10)

Jawab:
x + y + z = - 15
U(3) + U(4) + U(5) = -15
(a + (3-1)b) + (a + (4-1)b) + (a+ (5-1)b) = -15
(a + 2b) + (a + 3b) + (a + 4b) = - 15
3a + 10b = -15...(1)

x + y + z = -15
(a + 5b) + (a + 3b) + (a + 4b) = -15
3a + 12b = -15

3a + 12b = -15
3a + 10b = -15
__________ _
2b = 0
b = 0
3a = -15
a = -15/3
a = -5

U(10)
= a + (10 - 1) b
= -5 + 9(0)
= -5

z - y = y - x
z + x = 2y

x + y + z = -15
2y + y = -15
3y = -15
y = -5

z - y = (a + b) - a
z - (-5) = b
z = b - 5

y - x = (a + b) - a
-5 - x = b
x = -b - 5

(a + 5b)/(a + b) = (a + b)/a
a(a + 5b) = (a + b)(a + b)
a^2 + 5ab = a^2 + 2ab + b^2
a^2 + 2ab + 3ab = a^2 + 2ab + b^2
b^2 = 3ab
b = 3a

maka, barisannya aritmetikanya akan menjadi :

a, a+b, x, y, z
a, a+3a, -b-5, -5, b-5
a, 4a, -3a-5, -5, 3a+5

-3a - 5 = 4a - a
-3a - 5 = 3a
6a = -5
a = -5/6

beda = 4a - a = 3a
3a = (-5/6)*3 = -5/2

U10 = a + (n - 1)b
= -5/6 + (10 - 1)(-5/2)
= -5/6 + 9(-5/2)
= -5/6 - 45/2
= -5/6 - 135/6
= -140/6
= -70/3