For a positive integer n, the rational number r satisfies
2n/n + 2n/(n + 1) = r*2n/n - n/(n - 1)
Express r in terms of n. (Your expression should be as simplified as possible.)
We can simplify the equation by finding a common denominator for all the fractions. The least common multiple of n, n+1, and n-1 is n(n+1)(n-1). Multiplying through the equation by n(n+1)(n-1), we get:
2(n+1)(n-1) + 2n(n-1) = r*2n(n-1) - n(n+1)
Simplifying the left side, we get:
2(n^2-1) + 2n^2(n-1) = r*2n(n-1) - n^2 - n
Expanding and collecting like terms on both sides, we get:
2n^2 - 2 + 2n^3 - 2n^2 = 2n^2r - n^2 - n
Simplifying further, we have:
2n^3 = 2n^2r - n
Dividing both sides by 2n, we get:
n^2 = nr - 1/2
Rearranging terms, we have:
nr = n^2 + 1/2
Therefore, the expression for r in terms of n is:
r = (n^2 + 1/2) / n
To express r in terms of n, let's simplify the given equation step by step.
Starting with the given equation:
2n/n + 2n/(n + 1) = r*2n/n - n/(n - 1)
Simplifying the fractions on both sides of the equation:
2 + 2n/(n + 1) = r*2 - n/(n - 1)
Combine the terms on the left side:
2n/(n + 1) = r*2 - (n - 2n)/(n - 1)
2n/(n + 1) = r*2 - (-n + 2n)/(n - 1)
2n/(n + 1) = r*2 - (n/(n - 1))
Now, let's find a common denominator for the fractions on the right side:
2n/(n + 1) = [(r*2)(n - 1) - n(n + 1)]/(n - 1)
Expanding the terms on the right side:
2n/(n + 1) = [2rn - 2r - n^2 - n]/(n - 1)
Since both sides have the same denominator, we can remove it:
2n = 2rn - 2r - n^2 - n
Rearranging the terms to isolate n:
2n + n^2 + n = 2rn - 2r
Combining like terms on the left side:
n^2 + 3n = 2rn - 2r
Now, let's factor out n on the left side:
n(n + 3) = 2rn - 2r
Dividing both sides by (n + 3):
n = [2rn - 2r]/(n + 3)
Now, we want to express r in terms of n. To do that, we can isolate r by cross-multiplying:
n(n + 3) = 2rn - 2r
n^2 + 3n = 2rn - 2r
n^2 + 3n + 2r = 2rn
Rearranging the terms to isolate r:
2r - 2rn = -n^2 - 3n
2r(1 - n) = -n^2 - 3n
r = (-n^2 - 3n)/(2(1 - n))
This is the expression for r in terms of n.
To find the expression for r in terms of n, we will first simplify the given equation step-by-step.
Step 1: Simplify the fractions on both sides of the equation.
2n/n = 2
2n/(n + 1) = 2n/(n + 1) (no simplification possible)
2n/n - n/(n - 1) = (2n - n)/(n - 1) = n/(n - 1)
So, the given equation becomes:
2 + 2n/(n + 1) = r * (2n/n - n/(n - 1))
Step 2: Distribute r on the right-hand side of the equation.
2 + 2n/(n + 1) = r * 2n/n - r * n/(n - 1)
Step 3: Simplify the fractions on the right-hand side of the equation.
2n/n = 2
n/(n - 1) = n/(n - 1)
So, the equation becomes:
2 + 2n/(n + 1) = r * 2 - r * n/(n - 1)
Step 4: Simplify the left-hand side of the equation.
2 + 2n/(n + 1) = 2(n + 1)/(n + 1) + 2n/(n + 1) = (2n + 4)/(n + 1)
So, the equation becomes:
(2n + 4)/(n + 1) = r * 2 - r * n/(n - 1)
Step 5: Multiply both sides of the equation by (n + 1) to eliminate the denominator on the left-hand side.
(n + 1) * (2n + 4)/(n + 1) = (n + 1) * (r * 2 - r * n/(n - 1))
This simplifies to:
2n + 4 = (n + 1) * (r * 2 - r * n/(n - 1))
Step 6: Expand the expression on the right-hand side of the equation.
2n + 4 = 2r * (n + 1) - (r * n/(n - 1)) * (n + 1)
Step 7: Simplify the equation by multiplying terms.
2n + 4 = 2rn + 2r - rn - r/(n - 1)
Step 8: Combine the like terms on the right-hand side.
2n + 4 = (2rn - rn) + (2r - r/(n - 1))
Simplifying this equation further is not possible, so the expression for r in terms of n is:
r = (2n + 4 - 2rn + rn + r/(n - 1))/(2 - 1/(n - 1))