Question

in a geometric sequence, t4 = 5x - 11, t5 = x + 5, and t6 = x - 1. Find the sum(s) of the infinite series if it (or they) exist(s).

Answer 1

the sum exists if |r| < 1

Answer 2

find the value of `x` that gives you |r| < 1

Answer 3

but how do i find x?

Answer 4

notice thatt the given terms are consecutive terms in a geometric sequence,
so the ratio of any two adjacent terms equals the common ratio \(r\)
:
\[\large r = \dfrac{x+5}{5x-11} = \dfrac{x-1}{x+5}\]

Answer 5

take the last two ratios and solve x :

\[\large (x+5)^2 = (5x-11)(x-1)\]

Answer 6

Assume that r = 1/2
Then we can write:
\[\frac{1}{2}(5x-11)=x+5\]
2.5x - 5.5 = x + 5
1.5x = 10.5
x = 7
When x = 7 the geometric sequence becomes:
24, 12, 6, ...............

Answer 7

I cannot get the method of @ganeshie8 to work.

Answer 8

ahhh omg you're a lifesaver xD THANK YOU SO MUCH :D

Answer 9

You're welcome :)

Answer 10

My bad. The method of @ganeshie8 works perfectly, and should get higher marks!

Answer 11

so either way, they both work?

Answer 12

Yes, they both work.