Question

the product of two consecutive natural numbers is greater than their sum by 155. find these numbers.
answer
the first number is (square). the second number is (square).

Explanation:

Step1: Define variables

Let the first natural number be \( x \), then the next consecutive natural number is \( x + 1 \).

Step2: Set up the equation

The product of the two numbers is \( x(x + 1) \), and their sum is \( x + (x + 1) \). According to the problem, the product is greater than the sum by 155, so we have the equation:
\[
x(x + 1) = x + (x + 1) + 155
\]

Step3: Simplify the equation

Expand the left side: \( x^2 + x \)
Simplify the right side: \( 2x + 1 + 155 = 2x + 156 \)
So the equation becomes:
\[
x^2 + x = 2x + 156
\]
Subtract \( 2x + 156 \) from both sides to get a quadratic equation:
\[
x^2 - x - 156 = 0
\]

Step4: Solve the quadratic equation

We can factor the quadratic equation. We need two numbers that multiply to -156 and add up to -1. The numbers are -13 and 12. So:
\[
(x - 13)(x + 12) = 0
\]
Setting each factor equal to zero gives \( x - 13 = 0 \) or \( x + 12 = 0 \). Solving these, we get \( x = 13 \) or \( x = -12 \). But since we are dealing with natural numbers, we discard \( x = -12 \).

Step5: Find the two numbers

If \( x = 13 \), then the next consecutive number is \( 13 + 1 = 14 \).

Answer:

The first number is \( 13 \). The second number is \( 14 \).