<s>[INST] Geometric sequence $a,b,c,d$ and arithmetic sequence $a, \frac{b}{2},\frac{c}{4}, d-70$ The first four terms, given in order, of a geometric sequence $a,b,c,d$ and arithmetic sequence $a, \frac{b}{2},\frac{c}{4}, d-70$, find the common ratio $r$ and the values of each $a,b,c,d$.
What I have tried:
$b=ar, c=ar^2,d=ar^3$, we have find the value of $b$ by calculating the mean from the values $1$ and $3$.$$ar=a+\frac{ar^2}{4} \implies 0 = a(r^2-4r+4a) \\0 = a(r-2)(r+2)$$
To calculate for $c$ we take the second and fourth term of the sequence.
$$\frac{ar^2}{2}=\frac{ar}{2}+ar^3-70 \implies 70 = a(r^3-\frac{r^2}{2}+\frac{r}{2})$$
However, this seems a little messy because it involves complex numbers - am I still on the right track?
 [/INST] There is a concealed "joke" in this problem which is obscured by choosing to solve directly for $ \ r \ \ . $  If we instead seek the common difference $ \ \Delta \ $ between the terms of the arithmetic sequence, we might start by writing those terms as $ \ a \ \ , \ \ a + \Delta \ = \ \frac{b}{2} \ \ , \ \ a \ + \ 2 \Delta \ = \ \frac{c}{4} \ \ , \ \ a \ + \ 3 \Delta \ = \ d - 70 \ \ . $   In the geometric sequence then, the common ratio between terms is found from
$$ r \ \ = \  \ \frac{2a \ + \ 2 \Delta}{a} \ \ = \ \ \frac{4a \ + \ 8 \Delta}{2a \ + \ 2 \Delta} \ \ = \ \ \frac{a \ + \ 3 \Delta \ + \ 70}{4a \ + \ 8 \Delta} \ \ . $$
Cross-multiplication of the first two ratios (assuming $ \ a \neq 0 \ $ and $ \ a \ \neq \ -\Delta \ , \ $ which in fact cannot work for both sequences simulatneously)    produces
$$  a·(4a + 8 \Delta) \ \ = \ \  (2a + 2 \Delta)^2 \ \ \Rightarrow \ \ 4a^2 \  + \  8a \Delta \ \ = \ \ 4a^2 \ + \ 8a \Delta \ + \ 4 \Delta^2 \ \ \Rightarrow \ \ 4 \Delta^2 \ = \ 0 \ \ (!) \ \ . $$
[Yeah, I didn't believe this the first time I ran through a solution either...]
Since $ \ a \ = \ 0 \ $ will not "work" in the arithmetic sequence, we must indeed have $ \ \Delta \ = \ 0 \ \ , $ which of course means that all of the arithmetic sequence terms are identical.  The equation for the geometric series ratio "collapses" to
$$ r \ \ = \  \ \frac{2a }{a} \ \ = \ \ \frac{4a }{2a } \ \ = \ \ 2 \ \ = \ \  \frac{a \ + \ 70}{4a } \ \ , $$
with the equality of the last two segments of the equation yielding $ \ 8a \ = \ a + 70 \ \Rightarrow \ a \ = \ 10 \ \ . $  Hence, the geometric sequence is $ \ 10 \ , \ 20 \ , \ 40 \ , \ 80 \ \ , $ making the arithmetic sequence
$ \ 10 \ \ , \ \  \frac{20}{2} \ \ , \ \  \frac{40}{4} \ \ , \ \  80 - 70 \ \ . $
(Very amusing, Problem-Poser Person...)
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