Answer

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**Hint:**We can see that these numbers are perfect squares of the first few natural numbers. To find out the missing number, we will try to establish a pattern in these numbers and then we will apply that pattern and find our missing number. We can try finding differences between the terms, checking if they are factors or multiples of one number or if they are perfect squares.

**Complete step by step answer:**

Here, we can see that the numbers are perfect squares. Thus we can write them as

\[\begin{align}

& \Rightarrow 49,16,25,36,9 \\

& {{7}^{2}},{{4}^{2}},{{5}^{2}},{{6}^{2}},{{3}^{2}} \\

\end{align}\]

We can observe that starting from the first number, i.e. 49, i.e. ${{7}^{2}}$ , every second number is the square of the last odd number.

If we separate them out, they will come out to be:

$\begin{align}

& 49,25,9 \\

& \Rightarrow {{7}^{2}},{{5}^{2}},{{3}^{2}} \\

\end{align}$

Now, if we start from the second number, i.e. 16, i.e. ${{4}^{2}}$ , we can see that every second number is the square of the next even number.

If we separate them out, they will come out to be:

$\begin{align}

& 16,36 \\

& \Rightarrow {{4}^{2}},{{6}^{2}} \\

\end{align}$

Now , we can see that the series we have has alternative squares of odd and even numbers, i.e. , it has been made as a merger of the two series and it starts from the odd number series.

Thus, we have:

${{7}^{2}},{{4}^{2}},{{5}^{2}},{{6}^{2}},{{3}^{2}},?$

As the series starts from odd, the next is square of an even number, the next to that is the square of an odd number and so on.

The last even number square in the series is the square of ‘6’ and the last odd number square in the series is the square of ‘3’.

Thus, the next even number square will be the square of ‘8’ and the next odd number square will be the square of ‘1’.

The last number in the series is 9, i.e. ${{3}^{2}}$ and it is the square of an odd number. Hence, the next number will be the square of an even number.

Thus the next number of the series will be:

$\Rightarrow {{8}^{2}}=64$

Therefore, 64 will be the next number in the series.

**So, the correct answer is “Option A”.**

**Note:**Be careful while finding the pattern in a series. The pattern should be regular and logical and it should be applied to every member of the series. Some students may try to decode the missing number with the help of the options given but it is in fact a waste of time in these kinds of questions as they tend to rather confuse the students instead of getting the right answer.

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