What will come in place of the question mark ( ?) in each of the following number series ?
`17" "9" "?" "16.5" "35" "90`

Options

A. 5
B. 15
C. 10
D. 20

Answer

To find the missing number in the series `17, 9, ?, 16.5, 35, 90`, we can analyze the pattern of the numbers step by step.

### Step-by-step Solution:

1. **Identify the first two numbers:**
- The first number is 17 and the second number is 9.
- The difference between them is \( 17 - 9 = 8 \).

2. **Determine the operation:**
- It seems that the series is decreasing from 17 to 9.
- We can observe that \( 9 = 17 - \frac{17}{2} + \frac{1}{2} \) which simplifies to \( 9 = 17 - 8.5 + 0.5 \).

3. **Find the missing number:**
- Now, we need to find the number that comes after 9 and before 16.5.
- The next number should be derived from 9. If we look at the next number, 16.5, we can see that it is increasing.
- If we multiply 9 by \( \frac{5}{3} \) (which is approximately 1.67), we can get close to 15.
- However, to find a simpler pattern, let's try adding a number to 9. If we add 1 to 9, we get 10.
- Now, if we multiply 10 by 1.5, we get \( 10 \times 1.5 = 15 \), which is still not 16.5.

4. **Test the addition of 1:**
- If we take 9 and add 1, we get 10.
- If we then multiply 10 by 1.5, we get \( 10 \times 1.5 = 15 \).
- Adding 1.5 to 15 gives us \( 15 + 1.5 = 16.5 \).

5. **Confirm the pattern:**
- Now, we can see that the pattern is:
- From 17 to 9, we subtract 8.
- From 9 to 10, we add 1.
- From 10 to 16.5, we multiply by 1.5 and then add 1.5.
- From 16.5 to 35, we can see that it is multiplied by 2 and added 2.
- From 35 to 90, we can see that it is multiplied by 3 and added 3.

Thus, the missing number in the series is **10**.