Question

The sum of three consecutive odd integers is $99$. What are the three numbers?

Answer 1

Step 1: Use the definition of arithmetic progression to find the three numbers.

We are given three consecutive odd numbers,

Let the three consecutive odd numbers be:$2\mathrm{n}+1,2\mathrm{n}+3,2\mathrm{n}+5$.

Now, the sum of these three terms is $99$.

On adding these terms the equation becomes:

$\begin{array}{rcl}2n+1+2n+3+2n+5& =& 99\\ & \Rightarrow & 6n+9=99\\ & \Rightarrow & 6n=90\\ & \Rightarrow & n=15\end{array}$

Step 2: Find the numbers

Now substituting the value of $\mathrm{n}$ in the above:

$$\begin{gathered} 2n+1=2\left(15\right)+1 \\ \Rightarrow 2n+1=30+1 \\ \Rightarrow 2n+1=31 \end{gathered}$$

Hence the next odd numbers will be $33,35$.

Therefore, the three odd numbers are $\mathbf{31}\mathbf{,}\mathbf{33}\mathbf{,}\mathbf{35}$.