If the sequence `(a_(n))` is in GP, such that `a_(4)//a_(6)=1//4 and a_(2)+a_(5)=216,` then `a_(1)` is equal to

Options

A. 12 or `(108)/(7)`
B. 10
C. 7 or `(54)/(7)`
D. none of these

Explanation

To solve the problem, we will follow these steps:

1. **Understand the properties of a Geometric Progression (GP)**: In a GP, each term can be expressed in terms of the first term \( a_1 \) and the common ratio \( r \). The \( n \)-th term is given by:
\[ a_n = a_1 \cdot r^{n-1} \]

2. **Set up the equations based on the given information**:
- From the first condition \( \frac{a_4}{a_6} = \frac{1}{4} \):
\[ \frac{a_1 \cdot r^3}{a_1 \cdot r^5} = \frac{1}{4} \]
Simplifying this gives:
\[ \frac{r^3}{r^5} = \frac{1}{4} \implies \frac{1}{r^2} = \frac{1}{4} \implies r^2 = 4 \implies r = 2 \text{ or } r = -2 \]

- From the second condition \( a_2 + a_5 = 216 \):
\[ a_1 \cdot r + a_1 \cdot r^4 = 216 \]
Factoring out \( a_1 \):
\[ a_1 (r + r^4) = 216 \]

3. **Calculate \( a_1 \) for both cases of \( r \)**:
- **Case 1**: When \( r = 2 \):
\[ r + r^4 = 2 + 2^4 = 2 + 16 = 18 \]
Therefore:
\[ a_1 \cdot 18 = 216 \implies a_1 = \frac{216}{18} = 12 \]

- **Case 2**: When \( r = -2 \):
\[ r + r^4 = -2 + (-2)^4 = -2 + 16 = 14 \]
Therefore:
\[ a_1 \cdot 14 = 216 \implies a_1 = \frac{216}{14} = \frac{108}{7} \]

4. **Conclusion**: The possible values for \( a_1 \) are \( 12 \) and \( \frac{108}{7} \).

Thus, the final answer is:
\[ a_1 = 12 \text{ or } a_1 = \frac{108}{7} \]