Let 3, a, b, c be in A.P. and 3, a – 1, b +1, c + 9 be in G.P. Then, the arithmetic mean of a, b and c is :

Options

A. `-4`
B. `-1`
C. 13
D. 11

Answer

To solve the problem step by step, we need to follow the definitions of Arithmetic Progression (A.P.) and Geometric Progression (G.P.) as given in the question.

### Step 1: Understanding A.P.
Given that \(3, a, b, c\) are in A.P., we know that the middle term is the average of the other two. Therefore, we can write:
\[ 2a = 3 + b \quad \text{(Equation 1)} \]
Also, since \(a, b, c\) are in A.P., we have:
\[ 2b = a + c \quad \text{(Equation 2)} \]

### Step 2: Understanding G.P.
Now, we know that \(3, a-1, b+1, c+9\) are in G.P. This means that the ratios of consecutive terms are equal. Thus, we can write:
\[ \frac{a-1}{3} = \frac{b+1}{a-1} = \frac{c+9}{b+1} \quad \text{(Equation 3)} \]

### Step 3: Solving Equation 1
From Equation 1, we can express \(b\) in terms of \(a\):
\[ b = 2a - 3 \quad \text{(Substituting in Equation 1)} \]

### Step 4: Substituting \(b\) in Equation 2
Now substituting \(b\) into Equation 2:
\[ 2(2a - 3) = a + c \]
This simplifies to:
\[ 4a - 6 = a + c \]
Rearranging gives us:
\[ c = 3a - 6 \quad \text{(Equation 4)} \]

### Step 5: Substituting \(a\) and \(b\) into Equation 3
Now we substitute \(a\) and \(b\) into Equation 3. We first substitute \(b = 2a - 3\) and \(c = 3a - 6\):
\[ \frac{a-1}{3} = \frac{(2a - 3) + 1}{a - 1} \]
This simplifies to:
\[ \frac{a-1}{3} = \frac{2a - 2}{a - 1} \]
Cross-multiplying gives:
\[ (a - 1)^2 = 3(2a - 2) \]
Expanding both sides:
\[ a^2 - 2a + 1 = 6a - 6 \]
Rearranging gives:
\[ a^2 - 8a + 7 = 0 \]

### Step 6: Solving the Quadratic Equation
Now we can solve this quadratic equation using the quadratic formula \(a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
\[ a = \frac{8 \pm \sqrt{(-8)^2 - 4 \cdot 1 \cdot 7}}{2 \cdot 1} \]
Calculating the discriminant:
\[ = \frac{8 \pm \sqrt{64 - 28}}{2} = \frac{8 \pm \sqrt{36}}{2} = \frac{8 \pm 6}{2} \]
Thus, we have two possible values for \(a\):
\[ a = \frac{14}{2} = 7 \quad \text{or} \quad a = \frac{2}{2} = 1 \]

### Step 7: Finding Corresponding Values of \(b\) and \(c\)
1. **If \(a = 7\)**:
- \(b = 2(7) - 3 = 14 - 3 = 11\)
- \(c = 3(7) - 6 = 21 - 6 = 15\)

2. **If \(a = 1\)**:
- \(b = 2(1) - 3 = 2 - 3 = -1\)
- \(c = 3(1) - 6 = 3 - 6 = -3\)

### Step 8: Calculating the Arithmetic Mean
Now we need to find the arithmetic mean of \(a, b, c\):
1. For \(a = 7, b = 11, c = 15\):
\[ \text{AM} = \frac{7 + 11 + 15}{3} = \frac{33}{3} = 11 \]

2. For \(a = 1, b = -1, c = -3\):
\[ \text{AM} = \frac{1 - 1 - 3}{3} = \frac{-3}{3} = -1 \]

Since the problem does not specify negative values, we take the positive solution.

### Final Answer
The arithmetic mean of \(a, b, c\) is:
\[ \boxed{11} \]