If `(1.5)^(30)=k` , then the value of `sum_((n=2))^(29)(1.5)^n` , is :

Options

A. `2k-3`
B. `k +1`
C. `2k +7`
D. `2k - 9/2`

Answer

To solve the problem, we need to find the value of the summation \( \sum_{n=2}^{29} (1.5)^n \) given that \( (1.5)^{30} = k \).

### Step-by-Step Solution:

1. **Define the Summation**:
Let \( S = \sum_{n=2}^{29} (1.5)^n \).

2. **Relate to a Larger Summation**:
We know that:
\[ \sum_{n=1}^{29} (1.5)^n = (1.5)^1 + (1.5)^2 + (1.5)^3 + \ldots + (1.5)^{29} \]
We can express this summation in terms of a geometric series.

3. **Use the Geometric Series Formula**:
The sum of a geometric series can be calculated using the formula:
\[ S_n = A \frac{r^n - 1}{r - 1} \]
where \( A \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.

For our case:
- \( A = 1.5 \)
- \( r = 1.5 \)
- The number of terms from \( n=1 \) to \( n=29 \) is \( 29 \).

Thus, we have:
\[ \sum_{n=1}^{29} (1.5)^n = 1.5 \frac{(1.5)^{29} - 1}{1.5 - 1} = 1.5 \frac{(1.5)^{29} - 1}{0.5} = 3 \left( (1.5)^{29} - 1 \right) \]

4. **Calculate the Total Sum**:
To find \( \sum_{n=1}^{29} (1.5)^n \), we can also express it as:
\[ \sum_{n=1}^{29} (1.5)^n = S + (1.5)^1 = S + 1.5 \]
Therefore, we can equate:
\[ S + 1.5 = 3 \left( (1.5)^{29} - 1 \right) \]

5. **Express \( (1.5)^{29} \) in terms of \( k \)**:
Since \( (1.5)^{30} = k \), we have:
\[ (1.5)^{29} = \frac{k}{1.5} \]

6. **Substitute Back**:
Substitute \( (1.5)^{29} \) into the equation:
\[ S + 1.5 = 3 \left( \frac{k}{1.5} - 1 \right) \]
Simplifying this gives:
\[ S + 1.5 = 2k - 3 \]

7. **Solve for \( S \)**:
Rearranging gives:
\[ S = 2k - 3 - 1.5 = 2k - 4.5 \]

8. **Final Result**:
Therefore, the value of \( \sum_{n=2}^{29} (1.5)^n \) is:
\[ S = 2k - 4.5 \]

### Final Answer:
\[ \sum_{n=2}^{29} (1.5)^n = 2k - 4.5 \]