Question #41113

Jul 30, 2015

This series can only be a geometric sequence if $x = \frac{1}{6}$, or to the nearest hundredth $x \approx 0.17$.

Explanation:

The general form of a geometric sequence is the following:
$a , a r , a {r}^{2} , a {r}^{3} , \ldots$
or more formally ${\left(a {r}^{n}\right)}_{n = 0}^{\infty}$.

Since we have the sequence $x , 2 x + 1 , 4 x + 10 , \ldots$, we can set $a = x$, so $x r = 2 x + 1$ and $x {r}^{2} = 4 x + 10$.

Dividing by $x$ gives $r = 2 + \frac{1}{x}$ and ${r}^{2} = 4 + \frac{10}{x}$. We can do this division without problems, since if $x = 0$, then the sequence would be constantly $0$, but $2 x + 1 = 2 \cdot 0 + 1 = 1 \ne 0$. Therefore we know for sure $x \ne 0$.

Since we have $r = 2 + \frac{1}{x}$, we know
${r}^{2} = {\left(2 + \frac{1}{x}\right)}^{2} = 4 + \frac{4}{x} + \frac{1}{x} ^ 2$.
Furthermore we found ${r}^{2} = 4 + \frac{10}{x}$, so this gives:
$4 + \frac{10}{x} = 4 + \frac{4}{x} + \frac{1}{x} ^ 2$, rearranging this gives:
$\frac{1}{x} ^ 2 - \frac{6}{x} = 0$, multiplying by ${x}^{2}$ gives:
$1 - 6 x = 0$, so $6 x = 1$.
From this we conclude $x = \frac{1}{6}$.

To the nearest hundredth this gives $x \approx 0.17$.

Jul 30, 2015

As Daan has said, if the sequence is to be geometric, we must have $x = \frac{1}{6} \approx 0.17$ 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:

$\frac{2 x + 1}{x} = \frac{4 x + 10}{2 x + 1}$

Solving this equation gets us $x = \frac{1}{6}$