Three numbers are in arithmetic progression. Their sum is 3 and the sum of their squares is 11. What are the three numbers? I got one of the answers which is 5,7,9. What are the other three numbers?

The three numbers are -1, 1, and 3

Explanation:

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,

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.

If d = -2, then,

We get the same terms, just in the reverse order.

The sequence overall is still the same when we write the numbers from smallest to largest.

So there's only one solution and it's {-1, 1, 3}

geometry: triangles

solvers

lessons

answers archive

call the larger triangle abc

then the smaller triangle cde is similar to abc (all angles are equal, aaa).  

ab is the side that is 36, and we wish to find de, the side corresponding to ab

area_cde = area_abc - area_cde --> area_abc / area_cde = 2

let x=length of de

ratio of areas = (ratio of sides)^2

+2+=+%2836%2fx%29%5e2+

+2+=+1296%2fx%5e2+

++x%5e2+=+1296%2f2+=+648+

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