The Fast Way to Find 3 Consecutive Numbers with a Given Sum

The Fast Way to Find 3 Consecutive Numbers with a Given Sum

Finding three consecutive numbers that sum to a given value is a common problem in mathematics, often encountered in various contexts such as arithmetic sequences and algebra. This problem can be efficiently solved using algebraic methods. Here, we will explore a quick and systematic way to find three consecutive integers whose sum equals a given number.

Step-by-Step Method:

1. Define the Consecutive Numbers: Let's denote the three consecutive integers as $x-\mathrm{1\; , }$$x$ , and $x+1$ . Here, $x$ represents the middle number in the sequence.

2. Set Up the Equation: The sum of these three numbers can be written as:

   $$(x-1) + x + (x+1)$$ 

3. Simplify the Equation: Simplify the left side of the equation by combining like terms:

   $$(x-1) + x + (x+1) = 3x$$ 
   

   So, the equation becomes:

   $$3x = \text{Given Sum}$$ 

4. Solve for $x$ : To find the value of $x$ , divide the given sum by 3:

   $$x = \frac{\text{Given Sum}}{3}$$

5.  Determine the Three Consecutive Numbers: Once $x$ is found, the three consecutive integers are:

   $$x-1, \quad x, \quad x+1$$ 

Example:

Suppose we want to find three consecutive integers whose sum is 48.

1. Set Up the Equation:

   $$3x = 48$$ 

2. Solve for $x$ :

   $$x = \frac{48}{3} = 16$$ 

3. Find the Consecutive Numbers: The three consecutive numbers are:

   $$16-1 = 15, \quad 16, \quad 16+1 = 17$$ 

So, the three consecutive integers whose sum is 48 are 15, 16, and 17.

This method provides a quick and straightforward way to find three consecutive integers with a given sum, leveraging basic algebraic principles to simplify and solve the problem efficiently.