`P=(1+(1)/(x))(1+(1)/(x+1))(1+(1)/(x+2))...(1+(1)/(x+20))`. What is the value of P if x=9?

Options

A. 1.5
B. 2.25
C. 2.75
D. 3.33

Answer

To find the value of \( P \) when \( x = 9 \), we start with the expression:

\[ P = \left(1 + \frac{1}{x}\right)\left(1 + \frac{1}{x+1}\right)\left(1 + \frac{1}{x+2}\right) \cdots \left(1 + \frac{1}{x+20}\right) \]

### Step 1: Substitute \( x = 9 \)

First, we substitute \( x \) with 9 in the expression:

\[ P = \left(1 + \frac{1}{9}\right)\left(1 + \frac{1}{10}\right)\left(1 + \frac{1}{11}\right) \cdots \left(1 + \frac{1}{29}\right) \]

### Step 2: Simplify Each Term

Now, we simplify each term:

\[ 1 + \frac{1}{9} = \frac{10}{9}, \quad 1 + \frac{1}{10} = \frac{11}{10}, \quad 1 + \frac{1}{11} = \frac{12}{11}, \quad \ldots, \quad 1 + \frac{1}{29} = \frac{30}{29} \]

So we can rewrite \( P \) as:

\[ P = \frac{10}{9} \cdot \frac{11}{10} \cdot \frac{12}{11} \cdots \frac{30}{29} \]

### Step 3: Observe the Cancellation

Notice that in this product, every numerator cancels with the denominator of the next fraction:

\[ P = \frac{30}{9} \]

### Step 4: Calculate the Final Value

Now, we simplify \( \frac{30}{9} \):

\[ P = \frac{30}{9} = \frac{10}{3} \approx 3.333 \]

### Final Answer

Thus, the value of \( P \) when \( x = 9 \) is:

\[ P = \frac{10}{3} \]