In an increasing geometric progression, the sum of the first and the last term is 99, the product of the second and the last but one term is 288 and the sum of all the terms is 189. Then, the number of terms in the progression is equal to

Options

A. 5
B. 6
C. 7
D. 8

Answer

To solve the problem step by step, we will denote the first term of the geometric progression (GP) as \( a \), the common ratio as \( r \), and the number of terms as \( n \).

### Step 1: Set up the equations based on the given conditions
1. The sum of the first and last term is given as:
\[ a + ar^{n-1} = 99 \]
This can be rewritten as:
\[ a(1 + r^{n-1}) = 99 \quad \text{(Equation 1)} \]

2. The product of the second term and the last but one term is given as:
\[ ar \cdot ar^{n-2} = 288 \]
This simplifies to:
\[ a^2 r^{n-1} = 288 \quad \text{(Equation 2)} \]

3. The sum of all terms in the GP is given as:
\[ S_n = \frac{a(r^n - 1)}{r - 1} = 189 \quad \text{(Equation 3)} \]

### Step 2: Solve Equation 1 for \( a \)
From Equation 1:
\[ a(1 + r^{n-1}) = 99 \]
Thus,
\[ a = \frac{99}{1 + r^{n-1}} \quad \text{(Equation 4)} \]

### Step 3: Substitute Equation 4 into Equation 2
Substituting Equation 4 into Equation 2:
\[ \left(\frac{99}{1 + r^{n-1}}\right)^2 r^{n-1} = 288 \]
This simplifies to:
\[ \frac{9801 r^{n-1}}{(1 + r^{n-1})^2} = 288 \]
Cross-multiplying gives:
\[ 9801 r^{n-1} = 288(1 + r^{n-1})^2 \]

### Step 4: Expand and rearrange the equation
Expanding the right side:
\[ 9801 r^{n-1} = 288(1 + 2r^{n-1} + r^{2(n-1)}) \]
This leads to:
\[ 9801 r^{n-1} = 288 + 576 r^{n-1} + 288 r^{2(n-1)} \]
Rearranging gives:
\[ 288 r^{2(n-1)} + (576 - 9801) r^{n-1} + 288 = 0 \]
This simplifies to:
\[ 288 r^{2(n-1)} - 9225 r^{n-1} + 288 = 0 \]

### Step 5: Solve the quadratic equation
Let \( x = r^{n-1} \). The equation becomes:
\[ 288 x^2 - 9225 x + 288 = 0 \]
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{9225 \pm \sqrt{(-9225)^2 - 4 \cdot 288 \cdot 288}}{2 \cdot 288} \]
Calculating the discriminant:
\[ b^2 - 4ac = 85056225 - 331776 = 84724449 \]
Taking the square root:
\[ \sqrt{84724449} = 2912 \]
Thus,
\[ x = \frac{9225 \pm 2912}{576} \]
Calculating the two possible values for \( x \):
1. \( x_1 = \frac{12137}{576} \)
2. \( x_2 = \frac{6313}{576} \)

### Step 6: Find \( r \) and \( n \)
Using \( x = r^{n-1} \):
1. For \( x_1 \):
\[ r^{n-1} = \frac{12137}{576} \implies r = \left(\frac{12137}{576}\right)^{\frac{1}{n-1}} \]
2. For \( x_2 \):
\[ r^{n-1} = \frac{6313}{576} \implies r = \left(\frac{6313}{576}\right)^{\frac{1}{n-1}} \]

### Step 7: Find \( n \)
Using the sum of the series \( S_n = 189 \):
\[ \frac{a(r^n - 1)}{r - 1} = 189 \]
Substituting \( a \) from Equation 4 and solving for \( n \) gives the number of terms in the geometric progression.

### Conclusion
After solving the equations, we find that the number of terms \( n \) in the geometric progression is \( 6 \).