Find x respectively y in two sequences sharing terms - An arithmetic and geometric problem

Question

Problem:

Two different sequences contains both a and b.

Sequence ..., -1, a, b, x,... is arithmetic.

Sequence ..., y, a, b; 12,5;... is geometric.

determine x and y.

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I have been asked by a member of the mathoverflow to post here.

I have been going at this problem for several hours and haven't made any, in my opinion, real progress.

I welcome any solution to the problem :)

Answer 1

Let the common ratio of arithmetic progression be $d$.

$d=a+1=b-a=x-b$

Let the common ratio of geometric progression be $r$.

$r= \frac{a}{y}= \frac{b}{a}= \frac{12}{b}= \frac{5}{12}$

which gives $b= \frac{144}{5} \implies a= \frac{1728}{25} \implies y= \frac{20736}{125}.$

Now, the value of $b-a$ and $a+1$ do not match. Therefore, the question seems to be incorrect.