Question

Question #41113

Answer 1

This series can only be a geometric sequence if `x=1/6`, or to the nearest hundredth `xapprox0.17`.

Explanation:

The general form of a geometric sequence is the following:
`a,ar,ar^2,ar^3,...`
or more formally `(ar^n)_(n=0)^oo`.

Since we have the sequence `x,2x+1,4x+10,...`, we can set `a=x`, so `xr=2x+1` and `xr^2=4x+10`.

Dividing by `x` gives `r=2+1/x` and `r^2=4+10/x`. We can do this division without problems, since if `x=0`, then the sequence would be constantly `0`, but `2x+1=2*0+1=1ne0`. Therefore we know for sure `xne0`.

Since we have `r=2+1/x`, we know
`r^2=(2+1/x)^2=4+4/x+1/x^2`.
Furthermore we found `r^2=4+10/x`, so this gives:
`4+10/x=4+4/x+1/x^2`, rearranging this gives:
`1/x^2-6/x=0`, multiplying by `x^2` gives:
`1-6x=0`, so `6x=1`.
From this we conclude `x=1/6`.

To the nearest hundredth this gives `xapprox0.17`.

Answer 2

As Daan has said, if the sequence is to be geometric, we must have `x=1/6 ~~ 0.17` Here is one way to see that:

Explanation:

In a geometric sequence, the terms have a common ratio.

So, if this sequence is to be geometric, we must have:

`(2x+1)/x = (4x+10)/(2x+1)`

Solving this equation gets us `x = 1/6`