The sum of three consecutive terms of an arithmetic sequence is 27, and the sum of their square is 293. What is the absolute difference between the greatest and the least of these three numbers in the arthritic sequence?
Let
x -----> first consecutive term
so
x+d ----> second consecutive term
x+2d ----> third consecutive term
where
d -----> common factor
we have that
x+(x+d)+(x+2d)=27
3x+3d=27------> simplify -----> x+d=9 -----> equation 1
the sum of their square is 293
so
x^2+(x+d)^2+(x+2d)^2=293
x^2+(x^2+2xd+d^2)+(x^2+4xd+d^2)=293 ---------> equation 2
Solve the system by graphing
see the attached figure
the solution is the point (4,5)
so
x=4
d=5
therefore
the first term is 4
the second term is 4+5=9
the third term is 4+2(5)=14
the difference is 14-4=10