Question

If the sum of the three consecutive odd numbers is a perfect square between $200$ and $400$, then the root of this sum is:

• A. $15$
• B. $16$
• C. $18$
• D. $19$

Answer 1

The correct option is A

$15$

Solution:

Step 1: Finding the expression for the perfect square between $\mathbf{200}$ and $\mathbf{400}$ obeying the given condition:

The sum of the three consecutive odd numbers is a perfect square between $200$ and $400$,

Let, the three consecutive numbers be $x,x+2,x+4$.

Also,

$$\begin{gathered} 200<x+(x+2)+(x+4)<400 \\ \Rightarrow 200<(3x+6)<400 \\ \Rightarrow 200<3(x+2)<400 \end{gathered}$$

So, the required perfect square should be the multiple of $3$.

Step 2: Finding the perfect square between $200$ and $400$:

Perfect squares lying between $200\mathrm{and}400\mathrm{are}225,256,289,324,361.$

Among those squares, $225$ and $324$ are the multiples of $3$. But, sum of three consecutive odd numbers is always an odd number. So, the sum of the three consecutive odd numbers that is a perfect square between $200\mathrm{and}400$ is $225$.

Therefore, root of this sum is $\sqrt{225}=15$.

Final answer: Hence, option(A) is correct.