110, 90, 72, 56, ?

Options

A. 45
B. 42
C. 35
D. 30

Explanation

To solve the problem step by step, we will analyze the given series: 110, 90, 72, 56, ?

### Step 1: Identify the pattern in the series
We start by looking at the numbers in the series and trying to find a relationship between them.

### Step 2: Check if the numbers relate to perfect squares
We notice that the numbers are close to perfect squares. Let's calculate the squares of integers close to the numbers in the series:
- \(11^2 = 121\)
- \(10^2 = 100\)
- \(9^2 = 81\)
- \(8^2 = 64\)

### Step 3: Find the differences
Now, we will subtract the integers from their squares to see how they relate to the series:
- For 110: \(121 - 11 = 110\)
- For 90: \(100 - 10 = 90\)
- For 72: \(81 - 9 = 72\)
- For 56: \(64 - 8 = 56\)

### Step 4: Establish the pattern
From the above calculations, we can see that each term in the series can be represented as:
- \(n^2 - n\), where \(n\) is the integer starting from 11 and decreasing by 1 for each subsequent term.

### Step 5: Calculate the next term
To find the next term in the series, we will use \(n = 7\):
- \(7^2 - 7 = 49 - 7 = 42\)

### Conclusion
Thus, the missing number in the series is **42**.