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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 x1 , xx , and x+1x+1 . Here, xx represents the middle number in the sequence.

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

    (x1)+x+(x+1)(x-1) + x + (x+1) 
  3. Simplify the Equation: Simplify the left side of the equation by combining like terms:

    (x1)+x+(x+1)=3x(x-1) + x + (x+1) = 3x 

    So, the equation becomes:

    3x=Given Sum3x = \text{Given Sum} 
  4. Solve for xx : To find the value of xx , divide the given sum by 3:

    x=Given Sum3x = \frac{\text{Given Sum}}{3}
  5.  Determine the Three Consecutive Numbers: Once xx is found, the three consecutive integers are:

    x1,x,x+1x-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=483x = 48 
  2. Solve for xx :

    x=483=16x = \frac{48}{3} = 16 
  3. Find the Consecutive Numbers: The three consecutive numbers are:

    161=15,16,16+1=1716-1 = 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.

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