Insert the missing number in the given series:

• 0, 4, 18, 48, ?, 180

(a) 58

(b) 68

(c) 84

(d) 100

To find the missing number in the series 0, 4, 18, 48, ?, 180, we first look for a pattern in the numbers provided.

Identifying the Pattern

Let's examine the differences between consecutive terms:

• 4 - 0 = 4
• 18 - 4 = 14
• 48 - 18 = 30
• ? - 48
• 180 - ?

Calculating the Differences

The differences we have so far are:

• 4
• 14
• 30

Next, let's find the differences of these differences:

• 14 - 4 = 10
• 30 - 14 = 16

Continuing the Pattern

The second differences are 10 and 16, which increase by 6. If we continue this pattern, the next difference should be:

• 16 + 6 = 22

Now, we can find the next first difference:

• 30 + 22 = 52

Finding the Missing Number

Now, we can add this to the last known term in the series:

• 48 + 52 = 100

Thus, the missing number is 100.

Final Answer

The correct option is (d) 100.