Consider an `A. P .a_1,a_2,a_3,.....` such that `a_3+a_5+a_8 =11and a_4+a_2=-2` then the value of `a_1+a_6+a_7` is.....

Options

A. -8
B. 5
C. 7
D. 9

Explanation

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

### Step 1: Understand the terms of A.P.
The nth term of an A.P. can be expressed as:
\[ a_n = a_1 + (n-1)d \]
where \( a_1 \) is the first term and \( d \) is the common difference.

### Step 2: Write down the given equations.
From the problem, we have:
1. \( a_3 + a_5 + a_8 = 11 \)
2. \( a_4 + a_2 = -2 \)

### Step 3: Express the terms in terms of \( a_1 \) and \( d \).
Using the formula for the nth term:
- \( a_3 = a_1 + 2d \)
- \( a_5 = a_1 + 4d \)
- \( a_8 = a_1 + 7d \)

Substituting these into the first equation:
\[ (a_1 + 2d) + (a_1 + 4d) + (a_1 + 7d) = 11 \]
This simplifies to:
\[ 3a_1 + 13d = 11 \quad \text{(Equation 1)} \]

### Step 4: Express the second equation.
For the second equation:
- \( a_4 = a_1 + 3d \)
- \( a_2 = a_1 + d \)

Substituting these into the second equation:
\[ (a_1 + 3d) + (a_1 + d) = -2 \]
This simplifies to:
\[ 2a_1 + 4d = -2 \]
Dividing the entire equation by 2 gives:
\[ a_1 + 2d = -1 \quad \text{(Equation 2)} \]

### Step 5: Solve the equations.
Now we have two equations:
1. \( 3a_1 + 13d = 11 \)
2. \( a_1 + 2d = -1 \)

From Equation 2, we can express \( a_1 \) in terms of \( d \):
\[ a_1 = -1 - 2d \]

### Step 6: Substitute \( a_1 \) in Equation 1.
Substituting \( a_1 \) into Equation 1:
\[ 3(-1 - 2d) + 13d = 11 \]
This simplifies to:
\[ -3 - 6d + 13d = 11 \]
\[ 7d - 3 = 11 \]
Adding 3 to both sides:
\[ 7d = 14 \]
Dividing by 7:
\[ d = 2 \]

### Step 7: Find \( a_1 \).
Now substitute \( d = 2 \) back into Equation 2:
\[ a_1 + 2(2) = -1 \]
\[ a_1 + 4 = -1 \]
Subtracting 4 from both sides:
\[ a_1 = -5 \]

### Step 8: Calculate \( a_1 + a_6 + a_7 \).
Now we need to find \( a_1 + a_6 + a_7 \):
- \( a_6 = a_1 + 5d = -5 + 5(2) = -5 + 10 = 5 \)
- \( a_7 = a_1 + 6d = -5 + 6(2) = -5 + 12 = 7 \)

Now, calculate:
\[ a_1 + a_6 + a_7 = -5 + 5 + 7 = 7 \]

### Final Answer:
The value of \( a_1 + a_6 + a_7 \) is \( 7 \).

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