What will come in place of question mark(?) in the following number series?
5, 9 ,25, 61 ,125 ?

Options

A. 175
B. 150
C. 250
D. 225

Answer

To find the missing number in the series: 5, 9, 25, 61, 125, ?, we will analyze the differences between the consecutive terms and identify a pattern.

### Step 1: Identify the differences between consecutive terms
- The first term is 5.
- The second term is 9.
- The difference between the second and first term: 9 - 5 = 4.

- The third term is 25.
- The difference between the third and second term: 25 - 9 = 16.

- The fourth term is 61.
- The difference between the fourth and third term: 61 - 25 = 36.

- The fifth term is 125.
- The difference between the fifth and fourth term: 125 - 61 = 64.

Now we have the differences:
- 4, 16, 36, 64

### Step 2: Analyze the differences
Next, we notice that these differences themselves are perfect squares:
- 4 = 2²
- 16 = 4²
- 36 = 6²
- 64 = 8²

The pattern in the differences is that they are squares of even numbers:
- 2, 4, 6, 8

### Step 3: Determine the next difference
Following the pattern, the next even number after 8 is 10. Therefore, the next difference should be:
- 10² = 100

### Step 4: Calculate the next term in the series
Now, we add this difference to the last known term in the series:
- Last term = 125
- Next difference = 100

So, the next term will be:
- 125 + 100 = 225

### Conclusion
The missing term in the series is 225.

### Final Answer
The answer is 225.

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