Consider an `A.P. a_(1), a_(2), "……"a_(n), "……."` and the G.P. `b_(1),b_(2)"…..", b_(n),"….."` such that `a_(1) = b_(1)= 1, a_(9) = b_(9)` and `sum_(r=1)^(9) a_(r) = 369`, then

Options

A. `b_(6) = 27`
B. `b_(7) = 27`
C. `b_(8) = 81`
D. `b_(9) = 81`

Answer

To solve the problem, we need to analyze the given information about the arithmetic progression (A.P.) and the geometric progression (G.P.). Let's break it down step by step.

### Step 1: Understanding the A.P. and G.P.
We are given that:
- \( a_1 = b_1 = 1 \)
- \( a_9 = b_9 \)
- \( \sum_{r=1}^{9} a_r = 369 \)

### Step 2: Finding the common difference \( D \) of the A.P.
The general term of an A.P. can be expressed as:
\[ a_n = a_1 + (n-1)D \]
Thus, the first 9 terms of the A.P. are:
- \( a_1 = 1 \)
- \( a_2 = 1 + D \)
- \( a_3 = 1 + 2D \)
- ...
- \( a_9 = 1 + 8D \)

The sum of the first \( n \) terms of an A.P. is given by:
\[ S_n = \frac{n}{2} \left(2a_1 + (n-1)D\right) \]
For \( n = 9 \):
\[ S_9 = \frac{9}{2} \left(2 \cdot 1 + 8D\right) = 369 \]

### Step 3: Setting up the equation
Substituting the values into the sum formula:
\[ \frac{9}{2} (2 + 8D) = 369 \]
Multiplying both sides by 2:
\[ 9(2 + 8D) = 738 \]
Dividing by 9:
\[ 2 + 8D = 82 \]
Subtracting 2 from both sides:
\[ 8D = 80 \]
Dividing by 8:
\[ D = 10 \]

### Step 4: Finding \( a_9 \)
Now, we can find \( a_9 \):
\[ a_9 = 1 + 8D = 1 + 8 \cdot 10 = 1 + 80 = 81 \]

### Step 5: Finding the common ratio \( R \) of the G.P.
Since \( a_9 = b_9 \), we can express \( b_9 \) as:
\[ b_9 = b_1 R^{8} = 1 \cdot R^{8} = R^{8} \]
Setting this equal to \( a_9 \):
\[ R^{8} = 81 \]
Taking the eighth root:
\[ R = 81^{1/8} = 3^{4/8} = 3^{1/2} = \sqrt{3} \]

### Step 6: Finding \( b_6, b_7, b_8, b_9 \)
Now we can find the terms of the G.P.:
- \( b_6 = b_1 R^{5} = 1 \cdot (\sqrt{3})^{5} = 3^{5/2} = 9\sqrt{3} \)
- \( b_7 = b_1 R^{6} = 1 \cdot (\sqrt{3})^{6} = 3^{3} = 27 \)
- \( b_8 = b_1 R^{7} = 1 \cdot (\sqrt{3})^{7} = 3^{7/2} = 27\sqrt{3} \)
- \( b_9 = b_1 R^{8} = 1 \cdot (\sqrt{3})^{8} = 3^{4} = 81 \)

### Summary of Results
- \( b_6 = 9\sqrt{3} \)
- \( b_7 = 27 \)
- \( b_8 = 27\sqrt{3} \)
- \( b_9 = 81 \)

### Conclusion
Now we can check the options given in the problem to determine which are correct.