Finding first term of arithmetic sequence given first three terms and no common difference

Question

Assuming that the first three terms of an arithmetic sequence are $x, \frac{1}{x}, 1$ and $x<0$.

I seem to be unable to figure out what the first term is. I know that $a_n = a_1+d(n-1)$ but how do we work out the common difference in order to calculate $a_1$.

Is there anyway to calculate this recursively perhaps, given that we know the value of $a_3$.

I've tried manipulating the arithmetic formula above to figure this out but seem to be stuck. Can someone please point me in the right direction without flat-out giving the answer away?

Answer 1

By calculating the common difference in two different ways we have $$\frac1x-x=1-\frac1x$$ which simplifies to $$(x-1)(x+2)=0$$ and hence the first term is $x=-2$ as $x\lt0$.