In an arithmetic progression the `(p+1)^("th")` term is twice the `(q+1)^("th")` term. If its `(3p+1)^("th")` term is `lambda` times the`(p+q+1)^("th")` term, then `lambda` is equal to

Options

A. 2
B. `(1)/(2)`
C. 3
D. `(1)/(3)`

Answer

To solve the problem step by step, we will use the properties of an arithmetic progression (AP).

### Step 1: Understand the terms in an AP
In an arithmetic progression, the \( n^{th} \) term can be expressed as:
\[ a_n = a + (n-1)d \]
where \( a \) is the first term and \( d \) is the common difference.

### Step 2: Write the given conditions
According to the problem:
1. The \((p+1)^{th}\) term is twice the \((q+1)^{th}\) term.
2. The \((3p+1)^{th}\) term is \( \lambda \) times the \((p+q+1)^{th}\) term.

Using the formula for the \( n^{th} \) term, we can express these conditions mathematically.

### Step 3: Set up the first equation
From the first condition:
\[ a_{p+1} = 2 \cdot a_{q+1} \]
Substituting the formula for the terms:
\[ a + pd = 2(a + qd) \]
Expanding this:
\[ a + pd = 2a + 2qd \]
Rearranging gives:
\[ pd - 2qd = 2a - a \]
\[ pd - 2qd = a \]
Factoring out \( d \):
\[ (p - 2q)d = a \]

### Step 4: Set up the second equation
From the second condition:
\[ a_{3p+1} = \lambda \cdot a_{p+q+1} \]
Substituting the formula for the terms:
\[ a + 3pd = \lambda (a + (p+q)d) \]
Expanding this:
\[ a + 3pd = \lambda a + \lambda(p + q)d \]
Rearranging gives:
\[ a + 3pd - \lambda a - \lambda(p + q)d = 0 \]
Combining like terms:
\[ (1 - \lambda)a + (3p - \lambda(p + q))d = 0 \]

### Step 5: Solve for \( \lambda \)
For the equation to hold true, both coefficients must equal zero:
1. \( 1 - \lambda = 0 \) implies \( \lambda = 1 \)
2. \( 3p - \lambda(p + q) = 0 \)

From the first equation, we have \( \lambda = 1 \). Substituting \( \lambda = 1 \) into the second equation gives:
\[ 3p - (p + q) = 0 \]
This simplifies to:
\[ 3p - p - q = 0 \implies 2p = q \]

### Step 6: Substitute back to find \( \lambda \)
Now, substituting \( q = 2p \) back into the equation for \( \lambda \):
\[ \lambda = 2 \]

### Final Answer
Thus, the value of \( \lambda \) is:
\[ \lambda = 2 \]