if a b c and d are four positve increasing integers such that a,b c are in ap and b c d are in gp and D - a = 30, find the four numbers
#looks simple but still coudnt do it, neither could GPT 4 or gemini
blissful mountain
sterile vigilBOT
sterile vigilBOT
looks simple but still coudnt do it, neither could GPT 4 or gemini
limpid wyvern
You have that: $$\begin{cases} 2b=a+c \ c^2=bd \ d-a=30 \end{cases}$$
sacred bayBOT
Civil Service Pigeon
limpid wyvern
well use what I told you
you can find (a,b,c,d) in terms of only a and c
and then using c^2 = bd gets you the answer for free (note that ||you have a quadratic equation in c (or a)||)
Anyway, I gotta dip, but I think I gave a decent number of hints
blissful mountain
you can find (a,b,c,d) in terms of only a and c
I already tried
limpid wyvern
what did you do so far then
blissful mountain
Anyway, I gotta dip, but I think I gave a decent number of hints
Nothing new, already knew these
limpid wyvern
...
then you should say what you've done so far
b/c otherwise, ppl assume you're starting from square one
plus it saves time
blissful mountain
Sry I'm new here so didn't knew
limpid wyvern
Nothing new, already knew these
I presume you got to $$c=\frac{a+30 \pm \sqrt{9a^2+300a+900}}{4}$$
sacred bayBOT
Civil Service Pigeon
limpid wyvern
In this case, 9a^2+300a+900 needs to be a perfect square
blissful mountain
b/c otherwise, ppl assume you're starting from square one
Yes
limpid wyvern
So solve over the integers $$9a^2+300a+900=k^2$$
sacred bayBOT
Civil Service Pigeon
limpid wyvern
First, notice that a is a multiple of 3, so you can say that 9m^2 + 100m + 100 is a perfect square (where a = 3m)
To do this, consider:
- completing the square
- ||difference of squares||
you should get to ||r^2 = (3m + 50/3)^2 - 1600/9||
sterile vigilBOT
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