Question

Find three consecutive numbers whose sum is 234.

Answer 1

Hint: The three consecutive numbers differ by one. Use this fact to assign variables to the numbers and find the equation relating to the variables. Solve the equation to find the answer.

Complete step-by-step answer:
We need to find three consecutive numbers whose sum is 234.
We know that consecutive numbers are nearest neighbors to the number and which occur together.
For consecutive numbers, they differ by one number from the definition.
So, let us assume the three consecutive numbers to be x, y, and z in the same increasing order.
Then, using the fact that the difference between them is 1, we have as follows:
\[y - x = 1\]
\[z - y = 1\]
We get the value of y in terms of x.
$\Rightarrow$ \[y = x + 1............(1)\]
We find the value of z in terms of x.
$\Rightarrow$ \[z = y + 1\]
Using equation (1), we express z in terms of x.
$\Rightarrow$ \[z = x + 2.............(2)\]
It is given that the sum of the numbers is 234.
$\Rightarrow$ \[x + y + z = 234..........(3)\]
Substituting equation (1) and equation (2) in equation (3), we have:
$\Rightarrow$ \[x + x + 1 + x + 2 = 234\]
Simplifying, we get:
$\Rightarrow$ \[3x + 3 = 234\]
Solving for x, we get:
$\Rightarrow$ \[3x = 231\]
$\Rightarrow$ \[x = \dfrac{{231}}{3}\]
$\Rightarrow$ \[x = 77..........(4)\]
Substituting equation (4) in equation (1), we get the value of y.
$\Rightarrow$ \[y = 78\]
Substituting equation (4) in equation (2), we get the value of z.
$\Rightarrow$ \[z = 79\]
Hence, the three consecutive numbers are 77, 78, and 79.

Note: We can also choose the three consecutive even numbers as x-1, x, and x+1 so that the sum is 3x. The only necessity is that they need to differ by 1 each.