The average of the squares of seven consecutives integers is 53. The average of these integers is?
The tutor above, who solved it before me, suggested doing it a
harder way, by using
x, x+1, ... , x+6.
But it's much easier to let x be the middle integer instead of
the smallest integer. I see that the tutor below, who solved it
after I did, solved it the same way I did.
There are two solutions: 7 and -7. Here's how to
find them:
Let the 4th integer be x, then the 7 integers are
x-3, x-2, x-1, x, x+1, x+2, x+3
and their average is their sum over 7
x-3+x-2+x-1+x+x+1+x+2+x+3
—————————————————————————
7
Collecting terms:
7x
——
7
x
Their squares are:
(x-3)², (x-2)², (x-1)², x², (x+1)², (x+2)², (x+3)²
or
x²-6x+9, x²-4x+4, x²-2x+1, x², x²+x+1, x²+4x+4, x²+6x+9
The average of these squares is the sum of those squares
over 7
x²-6x+9+x²-4x+4+x²-2x+1+x²+x²+x+1+x²+4x+4+x²+6x+9
—————————————————————————————————————————————————
7
Collecting terms:
7x²+28
——————
7
Setting that equal to 53
7x²+28
—————— = 53
7
Multiply through by 7
7x²+28 = 371
7x² = 343
x² = 49
x = ±7
And since x is their average. the answer is ±7
So there are two solutions.
The integers are then either
-10, -9, -8, -7 , -6, -5, -4
or
4, 5, 6, 7, 8, 9, 10
Checking: the average of the square of either are
4² + 5² + 6² + 7² + 8² + 9² + 10²
—————————————————————————————————
7
16 + 25 + 36 + 49 + 64 + 81 + 100
—————————————————————————————————
7
371
———
7
53
So we're correct. There are two solutions.
Perhaps you were told only to find the positive
consecutive integers. If not, give both solutions.
Edwin