If for an `A.P. a_1,a_2,a_3,........,a_n,........a_1+a_3+a_5=-12 and a_1a_2a_3=8,` then the value of `a_2+a_4+a_6` equals

Options

A. `-12`
B. `-16`
C. `-18`
D. `-21`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Arithmetic Progression (A.P.)
Let the first term of the A.P. be \( a_1 = a \) and the common difference be \( r \). The terms can be expressed as:
- \( a_1 = a \)
- \( a_2 = a + r \)
- \( a_3 = a + 2r \)
- \( a_4 = a + 3r \)
- \( a_5 = a + 4r \)
- \( a_6 = a + 5r \)

### Step 2: Use the first condition
We know from the problem that:
\[ a_1 + a_3 + a_5 = -12 \]
Substituting the terms gives:
\[ a + (a + 2r) + (a + 4r) = -12 \]
Simplifying this:
\[ 3a + 6r = -12 \]
Dividing through by 3:
\[ a + 2r = -4 \quad \text{(Equation 1)} \]

### Step 3: Use the second condition
We also know:
\[ a_1 a_2 a_3 = 8 \]
Substituting the terms gives:
\[ a(a + r)(a + 2r) = 8 \]
Expanding this:
\[ a(a^2 + 3ar + 2r^2) = 8 \]
This simplifies to:
\[ a^3 + 3a^2r + 2ar^2 = 8 \quad \text{(Equation 2)} \]

### Step 4: Substitute \( a \) from Equation 1 into Equation 2
From Equation 1, we can express \( a \) as:
\[ a = -4 - 2r \]
Substituting this into Equation 2:
\[ (-4 - 2r)^3 + 3(-4 - 2r)^2r + 2(-4 - 2r)r^2 = 8 \]
Calculating \( (-4 - 2r)^3 \):
\[ = -64 - 48r - 24r^2 - 8r^3 \]
Calculating \( 3(-4 - 2r)^2r \):
\[ = 3(16 + 16r + 4r^2)r = 48r + 48r^2 + 12r^3 \]
Calculating \( 2(-4 - 2r)r^2 \):
\[ = -8r^2 - 4r^3 \]
Combining these:
\[ -64 - 48r - 24r^2 - 8r^3 + 48r + 48r^2 + 12r^3 - 8r^2 - 4r^3 = 8 \]
This simplifies to:
\[ -64 + 0r + 16r^2 = 8 \]
Thus:
\[ 16r^2 = 72 \implies r^2 = 4.5 \implies r = \pm \sqrt{4.5} \]

### Step 5: Find \( a \) and \( r \)
Using \( r = \sqrt{4.5} \) or \( r = -\sqrt{4.5} \), we can find \( a \):
From \( a + 2r = -4 \):
\[ a = -4 - 2r \]

### Step 6: Calculate \( a_2 + a_4 + a_6 \)
Now we need to find:
\[ a_2 + a_4 + a_6 = (a + r) + (a + 3r) + (a + 5r) = 3a + 9r \]
Substituting \( a = -4 - 2r \):
\[ = 3(-4 - 2r) + 9r = -12 - 6r + 9r = -12 + 3r \]

### Step 7: Substitute \( r \) back to find the final answer
Using \( r = \sqrt{4.5} \):
\[ = -12 + 3\sqrt{4.5} \]
If \( r = -\sqrt{4.5} \):
\[ = -12 - 3\sqrt{4.5} \]

### Final Result
The value of \( a_2 + a_4 + a_6 \) can be calculated based on the value of \( r \).