What will come in place of question mark (?) in the following series questions?
1,5,?,30,55,91

Options

A. A)13
B. B)10
C. C)9
D. D)14

Answer

To solve the series: 1, 5, ?, 30, 55, 91, we need to identify the pattern in the differences between the terms.

**Step 1: Identify the given terms in the series.**
The series is: 1, 5, ?, 30, 55, 91.

**Step 2: Calculate the differences between consecutive terms.**
- From 1 to 5: \(5 - 1 = 4\)
- From 5 to ?: Let’s denote this unknown term as \(K\).
- From \(K\) to 30: \(30 - K\)
- From 30 to 55: \(55 - 30 = 25\)
- From 55 to 91: \(91 - 55 = 36\)

So, we have the differences:
- First difference: 4
- Second difference: \(K - 5\)
- Third difference: \(30 - K\)
- Fourth difference: 25
- Fifth difference: 36

**Step 3: Analyze the differences.**
The differences we have are: 4, \(K - 5\), \(30 - K\), 25, 36.

**Step 4: Look for a pattern in the differences.**
Notice that:
- 4 is \(2^2\)
- 25 is \(5^2\)
- 36 is \(6^2\)

This suggests that the differences might be perfect squares. If we assume that the difference between 5 and \(K\) is \(3^2\) (which is 9), we can set up the equation:
\[ K - 5 = 9 \]
Thus,
\[ K = 5 + 9 = 14 \]

**Step 5: Verify the next difference.**
Now, we need to check if the difference from \(K\) (which we found to be 14) to 30 is \(4^2\) (which is 16):
\[ 30 - K = 30 - 14 = 16 \]
This is correct.

**Step 6: Confirm the differences.**
Now we have:
- From 1 to 5: 4 (which is \(2^2\))
- From 5 to 14: 9 (which is \(3^2\))
- From 14 to 30: 16 (which is \(4^2\))
- From 30 to 55: 25 (which is \(5^2\))
- From 55 to 91: 36 (which is \(6^2\))

The differences are indeed perfect squares, confirming our assumption.

**Final Answer:**
The value that comes in place of the question mark (?) is **14**.

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