p,q,r,s,t are first five terms of an A.P. such that P + r + t = -12 and p.q.r = 8. Find the first term of the above A.P. :

Options

A. 3
B. 2
C. 4
D. `-4`

Answer

To solve the problem, we need to find the first term \( P \) of the arithmetic progression (A.P.) given the conditions \( P + R + T = -12 \) and \( P \cdot Q \cdot R = 8 \).

### Step-by-Step Solution:

1. **Identify the terms of the A.P.**:
The first five terms of an A.P. can be expressed as:
- \( P = A - 2D \)
- \( Q = A - D \)
- \( R = A \)
- \( S = A + D \)
- \( T = A + 2D \)

2. **Use the first condition**:
From the problem, we know:
\[ P + R + T = -12 \]
Substituting the expressions for \( P \), \( R \), and \( T \):
\[ (A - 2D) + A + (A + 2D) = -12 \]
Simplifying this gives:
\[ 3A = -12 \]
Therefore:
\[ A = -4 \]

3. **Use the second condition**:
From the problem, we know:
\[ P \cdot Q \cdot R = 8 \]
Substituting the expressions for \( P \), \( Q \), and \( R \):
\[ (A - 2D) \cdot (A - D) \cdot A = 8 \]
Substituting \( A = -4 \):
\[ (-4 - 2D) \cdot (-4 - D) \cdot (-4) = 8 \]
This simplifies to:
\[ (-4)(-4 - 2D)(-4 - D) = 8 \]
Which can be rewritten as:
\[ 16(4 + 2D)(4 + D) = 8 \]
Dividing both sides by 16 gives:
\[ (4 + 2D)(4 + D) = \frac{1}{2} \]

4. **Expand and simplify**:
Expanding the left side:
\[ 16 + 8D + 4D + 2D^2 = \frac{1}{2} \]
Combining like terms:
\[ 2D^2 + 12D + 16 = \frac{1}{2} \]
Multiplying through by 2 to eliminate the fraction:
\[ 4D^2 + 24D + 32 = 1 \]
Rearranging gives:
\[ 4D^2 + 24D + 31 = 0 \]

5. **Solve the quadratic equation**:
Using the quadratic formula \( D = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 4 \), \( b = 24 \), and \( c = 31 \):
\[ D = \frac{-24 \pm \sqrt{24^2 - 4 \cdot 4 \cdot 31}}{2 \cdot 4} \]
Calculate the discriminant:
\[ 576 - 496 = 80 \]
Thus:
\[ D = \frac{-24 \pm \sqrt{80}}{8} \]
Simplifying \( \sqrt{80} = 4\sqrt{5} \):
\[ D = \frac{-24 \pm 4\sqrt{5}}{8} = \frac{-3 \pm \sqrt{5}}{2} \]

6. **Find \( P \)**:
Now, substituting \( D \) back to find \( P \):
\[ P = A - 2D = -4 - 2\left(\frac{-3 \pm \sqrt{5}}{2}\right) \]
This simplifies to:
\[ P = -4 + 3 \mp \sqrt{5} = -1 \mp \sqrt{5} \]

### Conclusion:
The first term \( P \) can take two values: \( -1 + \sqrt{5} \) or \( -1 - \sqrt{5} \).