We want to find the absolute difference between the greatest and the least of 3 consecutive terms of an arithmetic sequence that meet some given criteria. We will get that the difference is 3.88 or 12.88.
We know that the difference between any pair of two consecutive terms in an arithmetic sequence is a constant.
So if that difference is D, we can write 3 consecutive terms as:
A, A + D, A + 2*D
The sum must be equal to 27, then we have:
A + (A + D) + (A + 2*D) = 27
And the sum of their squares is equal to 293, then we have:
A^2 + (A + D)^2 + (A + 2*D)^2 = 293
If we simplify these two equations we get:
3*A + 2*D = 27
3*A^2 + 6*D*A + 5*D^2 = 293
To solve this, we just need to isolate one of the variables in the first equation and then replace that in the other one.
3*A = 27 - 2*D
A = (27 - 2*D)/3 = 9 - (2/3)*D
Replacing that on the other equation we get:
3*(9 - (2/3)*D)^2 + 6*(9 - (2/3)*D)*D + 5*D^2 = 293
Now we can solve this for D:
3*(81 - 2*9*(2/3)*D + D^2) + 54*D - 4*D^2 + 5*D^2 = 293
(243 - 293) + 18*D + 4*D^2 = 0
-50 + 18*D + 4*D^2 = 0
That is just a quadratic equation, the solutions are given by:
[tex]D = \frac{-18 \pm \sqrt{18^2 - 4*4*(-50)} }{2*4} \\\\D = \frac{-18 \pm 33.53}{8}[/tex]
Then we have two possible values for D, we take the positive one to get:
D = (-18 + 33.53)/8 = 1.94
And the difference between the greatest and least of these 3 numbers is twice the common difference, so is equal to 2*D = 2*1.94 = 3.88
If instead we take the other solution, we get:
D = (-18 - 33.53)/8 = -6.44
Then the absolute difference is -2*D = -2*-6.44 = 12.88
If you want to learn more, you can read:
https://brainly.com/question/25715593
