Suppose the decimal \(0.28\) is equal to \(\frac{1}{n}+\frac{2}{n^2}\), where \(n\) is negative. Find \(n\).
Solve for n:
1/n + 2/n^2 = 0.28
0.28 = 7/25:
1/n + 2/n^2 = 7/25
Bring 1/n + 2/n^2 together using the common denominator n^2:
(n + 2)/n^2 = 7/25
Cross multiply:
25 (n + 2) = 7 n^2
Expand out terms of the left hand side:
25 n + 50 = 7 n^2
Subtract 7 n^2 from both sides:
-7 n^2 + 25 n + 50 = 0
The left hand side factors into a product with three terms:
-(n - 5) (7 n + 10) = 0
Multiply both sides by -1:
(n - 5) (7 n + 10) = 0
Split into two equations:
n - 5 = 0 or 7 n + 10 = 0
Add 5 to both sides:
n = 5 or 7 n + 10 = 0
Subtract 10 from both sides:
n = 5 or 7 n = -10
Divide both sides by 7:
n = 5 or n = -10/7