Geometric Sequences and Series - Common Ratio Given & Two Non-Consecutive Terms

[markl77]

Hello,
this is the question I was given. For a geometric sequence, term 4 = 4x+8 and term 7 = x-4. If the common ratio is (1/2), what is the first term?

My initial work to do this question was t multiply (1/2) by term 4, which equals (2x+4), but if I go on with that common ratio it does not ever give me (x-4). I decided that I would have to figure out what x was before I could use the common ratio that was given.

To find x I looked at how term 7 - term 4 = 3, so therefore (1/2) * 3 would equal the amount I would have to multiply (4x+8) to get to (x-4).

1.5(4x+8)=(x-4)
6x+12=x-4
x=-3.2

So then I used -3.2 as x in the term (4x+8) going to the right to see if it would eventually equal (-7.2)

(4)(-3.2)+8=-4.8
-4.8*(1/2)=-2.4

And from there it just got smaller. I'm just wondering what I'm doing incorrectly in solving for x, and why in this case using (1/2) multiplied by the term (4x+8) doesn't eventually equal to (x-4)

Help is appreciated!

 

[Deleted member 4993]

Hello,
this is the question I was given. For a geometric sequence, term 4 = 4x+8 and term 7 = x-4. If the common ratio is (1/2), what is the first term?

My initial work to do this question was t multiply (1/2) by term 4, which equals (2x+4), but if I go on with that common ratio it does not ever give me (x-4). I decided that I would have to figure out what x was before I could use the common ratio that was given.

To find x I looked at how term 7 - term 4 = 3, so therefore (1/2) * 3 would equal the amount I would have to multiply (4x+8) to get to (x-4).

1.5(4x+8)=(x-4)
6x+12=x-4
x=-3.2

So then I used -3.2 as x in the term (4x+8) going to the right to see if it would eventually equal (-7.2)

(4)(-3.2)+8=-4.8
-4.8*(1/2)=-2.4

And from there it just got smaller. I'm just wondering what I'm doing incorrectly in solving for x, and why in this case using (1/2) multiplied by the term (4x+8) doesn't eventually equal to (x-4)

Help is appreciated!

let the first term of the GP be 'a' and common ratio 1/2. Then the sequence is:

a, a*(1/2), a*(1/2)², a*(1/2)³, a*(1/2)⁴, a*(1/2)⁵, a*(1/2)⁶, a*(1/2)^{7 .......} , then

a*(1/2)³= 4x + 8 → a = 32x + 64 and

a*(1/2)⁶ = x-4 → a = 64x - 256

32x + 64 = 64x - 256

x + 2 = 2x - 8

x = 10

Continue.......