**Finding the Consecutive Numbers**

Let's assume the two consecutive numbers as x and x+1.

**Setting up the Equation**

According to the problem, the reciprocals of these two numbers should sum up to 1/240. So, we can set up the equation as follows:

1/x + 1/(x+1) = 1/240

**Solving the Equation**

To solve this equation, we need to clear the denominators. Multiplying every term by the product of the denominators will eliminate the fractions.

240(x+1) + 240x = x(x+1)

Now, let's simplify the equation:

240x + 240 + 240x = x^2 + x

480x + 240 = x^2 + x

Rearranging the equation:

x^2 + x - 480x - 240 = 0

x^2 - 479x - 240 = 0

**Using the Quadratic Formula**

To solve this quadratic equation, we can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

Here, a = 1, b = -479, and c = -240.

x = (-(-479) ± √((-479)^2 - 4(1)(-240))) / (2(1))

x = (479 ± √(229441 + 960)) / 2

x = (479 ± √230401) / 2

**Calculating the Values of x**

Now, let's calculate the two possible values of x using the quadratic formula:

x₁ = (479 + √230401) / 2 ≈ 479.999

x₂ = (479 - √230401) / 2 ≈ -478.999

Since the numbers are consecutive, they cannot be negative. Therefore, the value of x = 479.999 is not valid.

**Finding the Consecutive Numbers**

So, the only valid value is x = 480. The two consecutive numbers are 480 and 481.

Therefore, the two consecutive numbers such that their reciprocals sum up to 1/240 are 480 and 481.