So the question is this:
Find the possible values of $x$ given the following consecutive terms of an arithmetic sequence:
\begin{eqnarray}
U_1 &=& x^4-8x^2-2007\\
U_2 &=& 2x^4-16x^2-4014\\
U_3 &=& 4x^4-84x^2-5482\\
\end{eqnarray}
I came up with an equation that I think you could solve to find $x$, but I'm really not sure at all. This is how I did it:
Let $d_1$ be the difference between the first two terms in the sequence. Let $d_2$ be the difference between the second and third term in the sequence.
\begin{eqnarray}
d_1 &=& U_2-U_1\\
d_2 &=& U_3-U_2
\end{eqnarray}
Since it is an arithmetic sequence, $d_1 = d_2$. It follows that $U_2-U_1=U_3-U_2$.
Working out the values of $U_2-U_1$ and $U_3-U_2$:
\begin{eqnarray}
U_3-U_2 &=& 2x^4-64x^2-1468\\
U_2-U_1 &=& x^4-8x^2-2007
\end{eqnarray}
That means that $2x^4-64x^2-1468 = x^4-8x^2-2007$. Is the working out right up to that point? If it is, you could simply just solve that equation for $x$, right?
Best Answer
Let's see:
$2x^4−68x^2−1468=x^4−8x^2−2007$
$y = x^2$
$2y^2-68y-1468=y^2-8y-2007$
$y^2-60y-539=0$
$(y-30)^2-539=900$
$(y-30)^2=1439$
$y-30=\pm 1439^{1/2}$
$y = 30\pm 1439^{1/2}$
$x = 1469^{1/2}$
$x = 1409^{1/2}$
Pressed enter accidentally up there and it looks weird as a comment.