First find the number in place of P in the following number series and then find the value of the expression given after the series.
`{:(188,186,P,174,158,126):}`
The value of `sqrt(P-13)` is

Options

A. `14.03`
B. `14.10`
C. `13.00`
D. `13.67`

Answer

To solve the problem step by step, we will first find the value of \( P \) in the given number series and then calculate the value of the expression \( \sqrt{P - 13} \).

### Step 1: Analyze the Series
The given series is:
\[ 188, 186, P, 174, 158, 126 \]

### Step 2: Find the Differences Between Consecutive Numbers
Let's find the differences between the consecutive numbers in the series:

- Difference between 188 and 186:
\[ 188 - 186 = 2 \]

- Difference between 186 and \( P \):
\[ P - 186 \]

- Difference between \( P \) and 174:
\[ P - 174 \]

- Difference between 174 and 158:
\[ 174 - 158 = 16 \]

- Difference between 158 and 126:
\[ 158 - 126 = 32 \]

### Step 3: Identify the Pattern
From the differences we calculated:
- The difference between 188 and 186 is 2.
- The difference between 174 and 158 is 16.
- The difference between 158 and 126 is 32.

We can see that the differences are increasing. The differences appear to follow a pattern of powers of 2:
- \( 2^1 = 2 \)
- \( 2^2 = 4 \) (this is the difference we need to find)
- \( 2^3 = 8 \) (this would be the difference between \( P \) and 174)
- \( 2^4 = 16 \)
- \( 2^5 = 32 \)

### Step 4: Calculate \( P \)
Since the difference between 186 and \( P \) should be \( 4 \) (which is \( 2^2 \)):
\[ P = 186 - 4 = 182 \]

### Step 5: Substitute \( P \) into the Expression
Now, we need to find the value of the expression \( \sqrt{P - 13} \):
\[ P - 13 = 182 - 13 = 169 \]

### Step 6: Calculate the Square Root
Now, we calculate the square root:
\[ \sqrt{169} = 13 \]

### Final Answer
Thus, the value of \( \sqrt{P - 13} \) is:
\[ \boxed{13} \]