Question 1055828
Since it is about 4 integers, consider divisibility.
{{{945=3^3*5*7}}} is the factorization for 945.
You notice that 3, 5, and 7 are factors of 945,
and the form an arithmetic sequence!
If those were 3 of the 4 factors whose product is 954,
The fourth factor would be
{{{945/(3*5*7)=3^3*5*7/(3*5*7)=3^3=9}}} .
So, {{{3*5*7*9=945}}} ,
and those 4 factors form an arithmetic sequence.
Of course, {{{-9}}} , {{{-7}}}, {{{-5}}}, and {{{-3}}} are another answer.
 
USING EQUATIONS:
{{{2x}}}= common difference of the arithmetic sequence.
{{{24/4=6}}}= average (mean) and median of thev4 numbers.
Then, the numbers are
{{{6-3x}}} , {{{6-x}}} , {{{6+x}}} , and {{{6+3x}}} .
The product is
{{{(6-3x)(6-x)(6+x)(6+3x)=945}}}
{{{(6^2-(3x)^2)(6^2-x^2)=945}}}
{{{(36-9x^3)(36-x^2)=945}}}
{{{9(4-x^3)(36-x^3)=945}}}
{{{(4-x^3)(36-x^2)=105}}}
{{{144-40x^2×x^4=105}}}
{{{x^4-40x+39=0}}}
{{{(x^2-1)(x^3-39)=0}}}
So,
EITHER,{{{x^2=1}}} ---> {{{system(x=1,"or",x=-1)}}} ,
which gives you the two arithmetic sequences found before,
OR {{{x^2=39}}} , which gives you
two arithmetic sequences of four (irrational) numbers,
asking up to 24, and with a product of 945.