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Question

The sum of the reciprocals of two consecutive positive integers is 1772
. Write an equation that can be used to find the two integers. What are the integers?

Answer 1

Let the two consecutive positive integers be x and x + 1.

The sum of their reciprocals is:
1/x + 1/(x + 1) = 1772

To solve this equation, we need to find a common denominator:
(x + 1)/x(x + 1) + x/(x(x + 1)) = 1772
(x + 1 + x)/(x(x + 1)) = 1772
(2x + 1)/(x^2 + x) = 1772

Now we can cross multiply to solve for x:
2x + 1 = 1772(x^2 + x)
2x + 1 = 1772x^2 + 1772x
0 = 1772x^2 + 1770x - 1

This is a quadratic equation, which can be solved using the quadratic formula:
x = [-b ± √(b^2 - 4ac)] / 2a
x = [-1770 ± √(1770^2 - 4(1772)(-1))] / 2(1772)
x = [-1770 ± √(3132900 + 7088)] / 3544
x = [-1770 ± √3139988] / 3544
x = [-1770 ± 1771.9] / 3544

This gives us two possible solutions:
x = 1.9 or x = 3542.9

Since x must be a positive integer, the only valid solution is x = 3542.9

Therefore, the two consecutive positive integers are 3542 and 3543.