the sum of the reciprocals of two consecutive positive numbers is 17/12 . Find the numbers
n and n+1
1/n + 1/(n+1) = 17/12
12(n+1) + 12 n = 17 n(n+1)
12 n + 12 + 12 n = 17 n^2 + 17 n
17 n^2 - 7 n - 12 = 0
hummm, suspect typo
maybe it's 7/12, since
1/3 + 1/4 = 7/12
Let's assume the two consecutive positive numbers as x and (x + 1).
According to the given problem, the sum of the reciprocals of these two numbers is 17/12:
1/x + 1/(x + 1) = 17/12
To solve the equation, we need to find a common denominator for the denominators of the fractions:
[(x + 1) + x] / (x(x + 1)) = 17/12
Simplifying the equation:
[2x + 1] / (x^2 + x) = 17/12
Cross-multiplying:
12(2x + 1) = 17(x^2 + x)
Expanding both sides:
24x + 12 = 17x^2 + 17x
Rearranging the equation:
17x^2 + (17x - 24x) + 12 = 0
Combining like terms:
17x^2 - 7x + 12 = 0
Factoring the quadratic equation:
(17x - 4)(x - 3) = 0
Setting each factor equal to zero:
17x - 4 = 0 -> x = 4/17
x - 3 = 0 -> x = 3
So, the two consecutive positive numbers are 4/17 and 3.
To find the two consecutive positive numbers, we can use algebraic equations. Let's assume that the first positive number is x, then the next consecutive positive number would be x + 1.
According to the given information, the sum of their reciprocals is equal to 17/12. So, we can set up the following equation:
1/x + 1/(x + 1) = 17/12
To solve this equation, we need to find a common denominator. In this case, the common denominator is 12x(x + 1). So we can multiply every term by 12x(x + 1) to clear the denominators:
12(x + 1) + 12x = 17x(x + 1)
Expanding and simplifying:
12x + 12 + 12x = 17x^2 + 17x
Combine like terms:
24x + 12 = 17x^2 + 17x
Rearrange to set the equation equal to zero:
17x^2 + 17x - 24x - 12 = 0
Combine like terms:
17x^2 - 7x - 12 = 0
At this point, we have a quadratic equation. We can solve this equation using factoring, completing the square, or the quadratic formula. In this case, factoring is the easiest method:
(17x + 9)(x - 4) = 0
Setting each factor equal to zero:
17x + 9 = 0, x - 4 = 0
Solving for x:
17x = -9, x = 4
x = -9/17, x = 4
Since we are looking for two consecutive positive numbers, we discard the negative solution. Therefore, the first positive number is x = 4 and the second consecutive positive number is x + 1 = 4 + 1 = 5.
So, the two consecutive positive numbers are 4 and 5.