If the sequence `{S_(n)}` is a geometric progression and `S_(2)S_(11)=S_(p)S_(8)`, then what is the value of p?

Options

A. 1
B. 3
C. 5
D. Cannot be determined

Answer

To solve the problem, we need to use the properties of a geometric progression (GP). Let's break down the steps:

### Step 1: Understand the terms in a GP
In a geometric progression, the n-th term can be expressed as:
\[ S_n = A \cdot R^{n-1} \]
where \( A \) is the first term and \( R \) is the common ratio.

### Step 2: Write expressions for the terms involved
We need to find \( S_2 \), \( S_{11} \), and \( S_8 \):
- \( S_2 = A \cdot R^{2-1} = A \cdot R^1 = A \cdot R \)
- \( S_{11} = A \cdot R^{11-1} = A \cdot R^{10} \)
- \( S_8 = A \cdot R^{8-1} = A \cdot R^7 \)

### Step 3: Substitute into the equation
We are given the equation:
\[ S_2 \cdot S_{11} = S_p \cdot S_8 \]
Substituting the expressions we derived:
\[ (A \cdot R) \cdot (A \cdot R^{10}) = S_p \cdot (A \cdot R^7) \]
This simplifies to:
\[ A^2 \cdot R^{11} = S_p \cdot (A \cdot R^7) \]

### Step 4: Solve for \( S_p \)
Rearranging the equation gives:
\[ S_p = \frac{A^2 \cdot R^{11}}{A \cdot R^7} \]
This simplifies to:
\[ S_p = A \cdot R^{11-7} = A \cdot R^4 \]

### Step 5: Identify the value of \( p \)
From the expression for \( S_p \), we can see that:
\[ S_p = A \cdot R^{p-1} \]
Setting the two expressions for \( S_p \) equal gives:
\[ A \cdot R^{p-1} = A \cdot R^4 \]
Since \( A \) is not zero, we can divide both sides by \( A \):
\[ R^{p-1} = R^4 \]
This implies:
\[ p - 1 = 4 \]
Thus, solving for \( p \):
\[ p = 4 + 1 = 5 \]

### Final Answer
The value of \( p \) is \( 5 \).

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