geometric sequence

consider the geometric sequence 1.5，－3，6，－12，... How many negative terms are
greater than －6000?
Sol
a1=－3，a2=－12
r=－12/(－3)=4
an=a1*r^(n－1)>－6000
(－3)*4^(n－1)>－6000
4^(n－1)<2000
4^5=1024
4^6=4096
n－1<=5
n<=6
6 negativeterms are greater than －6000

or
a1=－3
a2=－12
a3=－48
a4=－48*4=－192
a5=－192*4=－768
a6=－768*4=－3072>－6000
a7=－3072*4=－12288<－6000
6 negativeterms are greater than －6000

a(0) = 1.5; r = (-2); a(n) = a(0)r^(n-1)

a(n) .... 1.5 ... -3 ... 6 ........ -12 ...........
a(n) .....a(0)...a(0)r..a(0)r^2...a(0)r^3............a(0)r^(n-1)
n .........1.......2....... 3..........4......................n

let -6000 = (1.5)(-2)^(n-1)
6000 = -(1.5)(-1)^(n-1)(2)^(n-1)
6000 = (1.5)(-1)^n(2)^(n-1)

forget the (-1)^n for a moment

6000 = (1.5)(2)^(n-1)
6000 / 1.5 = (2)^(n-1)
4000 = (2)^(n-1)
log(4000) = (n-1)log(2)
n-1 = 11.96

n is approximately 13

Sub into a(0)r^(n-1)

(1.5)(-2)^(13-1) must > 0 since (-2)^12 > 0

Try n=14 (1.5)(-2)^13; it gives - (1.5)(2)^13 = -12288

Try n=12 (1.5)(-2)^11; it gives - (1.5)(2)^11 = -3072

So, the sequence is

a(n) .... 1.5 ... -3 ... 6 ........ -12 ............... -3072 ... 6144 .... -12288
a(n) .....a(0)...a(0)r..a(0)r^2...a(0)r^3............
n .........1.......2....... 3..........4.....................12........13...........14

Terms with n = 2, 4, 6, 8, 10, 12 are negative terms > -6000