If `x = (1*3)/(3*6)+(1*3*5)/(3*6*9)+(1*3*5*7)/(3*6*9*12)+. . . ` to infinite terms, then `9x^(2) + 24 x = `

Options

A. 11
B. 21
C. 31
D. 41

Answer

To solve the problem step by step, we will analyze the series and derive the value of \( x \) first, and then use that to find \( 9x^2 + 24x \).

### Step 1: Define the series
The series is given as:
\[ x = \frac{1 \cdot 3}{3 \cdot 6} + \frac{1 \cdot 3 \cdot 5}{3 \cdot 6 \cdot 9} + \frac{1 \cdot 3 \cdot 5 \cdot 7}{3 \cdot 6 \cdot 9 \cdot 12} + \ldots \]

### Step 2: Rewrite the terms
Notice that the general term can be expressed as:
\[ \frac{(2n-1)!!}{(3n)!!} \]
where \( n!! \) denotes the double factorial. The double factorial of an odd number can be expressed as:
\[ (2n-1)!! = \frac{(2n)!}{2^n n!} \]
and for the even numbers:
\[ (3n)!! = 3^n n! \]

### Step 3: Recognize the pattern
The series can be simplified using the binomial series expansion. The series resembles the expansion of:
\[ (1 - t)^{-1/2} = \sum_{n=0}^{\infty} \frac{(2n)!}{(n!)^2} \frac{t^n}{4^n} \]
for \( t = \frac{1}{3} \).

### Step 4: Find the value of \( x \)
Using the series expansion, we can write:
\[ x = \frac{1}{3} \sum_{n=0}^{\infty} \frac{(2n)!}{(n!)^2} \left(\frac{1}{3}\right)^n \]
This is equal to:
\[ x = \frac{1}{3} (1 - \frac{1}{3})^{-1/2} = \frac{1}{3} \left(\frac{2}{3}\right)^{-1/2} = \frac{1}{3} \cdot \sqrt{\frac{9}{2}} = \frac{3}{\sqrt{2} \cdot 3} = \frac{1}{\sqrt{2}} \]

### Step 5: Substitute \( x \) into the expression
Now we need to find \( 9x^2 + 24x \):
\[ x^2 = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \]
\[ 9x^2 = 9 \cdot \frac{1}{2} = \frac{9}{2} \]
\[ 24x = 24 \cdot \frac{1}{\sqrt{2}} = \frac{24\sqrt{2}}{2} = 12\sqrt{2} \]

### Step 6: Combine the results
Now we combine:
\[ 9x^2 + 24x = \frac{9}{2} + 12\sqrt{2} \]

### Step 7: Solve for the final value
To find a numerical approximation, we can calculate \( 12\sqrt{2} \):
\[ \sqrt{2} \approx 1.414 \Rightarrow 12\sqrt{2} \approx 12 \times 1.414 \approx 16.968 \]
Thus:
\[ 9x^2 + 24x \approx \frac{9}{2} + 16.968 \approx 4.5 + 16.968 \approx 21.468 \]

### Final Result
The final value of \( 9x^2 + 24x \) is approximately \( 21 \).