SOLUTION: Given that four integers are consecutive terms of an arithmetic sequence with sum 24 and product 945, what is the biggest of these four integers?

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Question 1055828: Given that four integers are consecutive terms of an arithmetic sequence with sum 24 and product 945, what is the biggest of these four integers?
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
Since it is about 4 integers, consider divisibility.
945=3%5E3%2A5%2A7 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%2F%283%2A5%2A7%29=3%5E3%2A5%2A7%2F%283%2A5%2A7%29=3%5E3=9 .
So, 3%2A5%2A7%2A9=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%2F4=6= average (mean) and median of thev4 numbers.
Then, the numbers are
6-3x , 6-x , 6%2Bx , and 6%2B3x .
The product is
%286-3x%29%286-x%29%286%2Bx%29%286%2B3x%29=945
%286%5E2-%283x%29%5E2%29%286%5E2-x%5E2%29=945
%2836-9x%5E3%29%2836-x%5E2%29=945
9%284-x%5E3%29%2836-x%5E3%29=945
%284-x%5E3%29%2836-x%5E2%29=105
144-40x%5E2%D7x%5E4=105
x%5E4-40x%2B39=0
%28x%5E2-1%29%28x%5E3-39%29=0
So,
EITHER,x%5E2=1 ---> system%28x=1%2C%22or%22%2Cx=-1%29 ,
which gives you the two arithmetic sequences found before,
OR x%5E2=39 , which gives you
two arithmetic sequences of four (irrational) numbers,
asking up to 24, and with a product of 945.