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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?

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  • $\begingroup$ I think I've got it figured out $d = -2$, could you please post the answer so I could confirm that I'm not totally off-base. $\endgroup$
    – Dustin
    Jul 11, 2020 at 14:13

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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 $x\lt0$.

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