If the sum of the first 15 terms of the series `((3)/(4))^(3)+(1(1)/(2))^(3)+(2(1)/(4))^(3)+3^(3)+(3(3)/(4))^(3)+...` is equal to 225k, then k is equal to

Options

A. 9
B. 27
C. 108
D. 54

Answer

To solve the problem, we need to find the value of \( k \) given that the sum of the first 15 terms of the series

\[ \left(\frac{3}{4}\right)^3 + \left(\frac{3}{2}\right)^3 + \left(\frac{9}{4}\right)^3 + 3^3 + \left(\frac{15}{4}\right)^3 + \ldots \]

is equal to \( 225k \).

### Step 1: Identify the pattern in the series

The terms of the series can be rewritten as:

- \( \left(\frac{3}{4}\right)^3 \)
- \( \left(\frac{3}{2}\right)^3 = \left(\frac{6}{4}\right)^3 \)
- \( \left(\frac{9}{4}\right)^3 = \left(\frac{9}{4}\right)^3 \)
- \( 3^3 = \left(\frac{12}{4}\right)^3 \)
- \( \left(\frac{15}{4}\right)^3 \)

We can see that the series can be expressed in terms of \( \left(\frac{3n}{4}\right)^3 \) for \( n = 1, 2, 3, \ldots, 15 \).

### Step 2: Rewrite the series

Thus, we can express the sum of the first 15 terms as:

\[ \sum_{n=1}^{15} \left(\frac{3n}{4}\right)^3 \]

### Step 3: Factor out \( \left(\frac{3}{4}\right)^3 \)

Factoring out \( \left(\frac{3}{4}\right)^3 \):

\[ \left(\frac{3}{4}\right)^3 \sum_{n=1}^{15} n^3 \]

### Step 4: Use the formula for the sum of cubes

The formula for the sum of the first \( n \) cubes is:

\[ \sum_{k=1}^{n} k^3 = \left(\frac{n(n+1)}{2}\right)^2 \]

For \( n = 15 \):

\[ \sum_{n=1}^{15} n^3 = \left(\frac{15 \cdot 16}{2}\right)^2 = (120)^2 = 14400 \]

### Step 5: Substitute back into the equation

Now substituting back into our expression:

\[ \left(\frac{3}{4}\right)^3 \cdot 14400 \]

Calculating \( \left(\frac{3}{4}\right)^3 \):

\[ \left(\frac{3}{4}\right)^3 = \frac{27}{64} \]

Thus, the sum becomes:

\[ \frac{27}{64} \cdot 14400 \]

### Step 6: Simplify the expression

Calculating \( \frac{27 \cdot 14400}{64} \):

First, simplify \( 14400 \div 64 \):

\[ 14400 \div 64 = 225 \]

Now, multiply by 27:

\[ 27 \cdot 225 = 6075 \]

### Step 7: Set the equation equal to \( 225k \)

We have:

\[ 6075 = 225k \]

### Step 8: Solve for \( k \)

Dividing both sides by 225:

\[ k = \frac{6075}{225} = 27 \]

Thus, the value of \( k \) is \( \boxed{27} \).

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