Answer: either {-1, 1, 3} or {3, 1, -1}
=====================================================
Step-by-step explanation:
- x = first number of the sequence
- x+d = second number
- (x+d)+d = x+2d = third number
- d = common difference
The three terms sum to
first + second + third = (x)+(x+d)+(x+2d) = 3x+3d
We're told this sum is 3, so,
3x+3d = 3
3(x+d) = 3
x+d = 3/3
x+d = 1
x = 1-d
The three terms x, x+d, x+2d have the squares of x^2, (x+d)^2, and (x+2d)^2 in that order.
Add up those squares and set the sum equal to 11
x^2 + (x+d)^2 + (x+2d)^2 = 11
Then plug in x = 1-d and solve for d.
x^2 + (x+d)^2 + (x+2d)^2 = 11
(1-d)^2 + (1-d+d)^2 + (1-d+2d)^2 = 11
(1-d)^2 + (1)^2 + (1+d)^2 = 11
(1-2d+d^2) + 1 + (1+2d+d^2) = 11
2d^2+3 = 11
2d^2 = 11-3
2d^2 = 8
d^2 = 8/2
d^2 = 4
d = sqrt(4) or d = -sqrt(4)
d = 2 or d = -2
----------------------
If d = 2, then,
- x = 1-d = 1-2 = -1
- x+d = -1+2 = 1
- x+2d = -1+2(2) = 3
One possible answer is to have the sequence -1, 1, 3
Summing the original terms gets us -1+1+3 = 3
Summing their squares gets us (-1)^2 + (1)^2 + (3)^2 = 1+1+9 = 11
We've confirmed this answer.
One possible answer is {-1, 1, 3}
--------------------
If d = -2, then,
- x = 1-d = 1-(-2) = 3
- x+d = 3-2 = 1
- x+2d = 3+2(-2) = -1
We get the same terms, just in the reverse order.
The other possible answer is {3, 1, -1}