The sum of the first three terms of the `G.P.` in which the difference between the second and the first term is `6` and the difference between the fourth and the third term is` 54`, is

Options

A. 39
B. -10.5
C. 27
D. -27

Answer

To solve the problem, we need to find the sum of the first three terms of a geometric progression (G.P.) given two conditions about the differences between certain terms. Let's denote the first term of the G.P. as \( a \) and the common ratio as \( r \). The first three terms of the G.P. can be expressed as \( a \), \( ar \), and \( ar^2 \).

### Step 1: Set up the equations based on the given conditions.

1. The difference between the second and first terms is given as \( 6 \):
\[ ar - a = 6 \]
Factoring out \( a \), we get:
\[ a(r - 1) = 6 \quad \text{(Equation 1)} \]

2. The difference between the fourth and third terms is given as \( 54 \):
\[ ar^3 - ar^2 = 54 \]
Factoring out \( ar^2 \), we get:
\[ ar^2(r - 1) = 54 \quad \text{(Equation 2)} \]

### Step 2: Solve the equations.

From Equation 1, we can express \( a \):
\[ a = \frac{6}{r - 1} \]

Now, substitute this expression for \( a \) into Equation 2:
\[ \frac{6}{r - 1} \cdot r^2(r - 1) = 54 \]
This simplifies to:
\[ 6r^2 = 54 \]
Dividing both sides by 6:
\[ r^2 = 9 \]
Taking the square root gives us:
\[ r = 3 \quad \text{or} \quad r = -3 \]

### Step 3: Find the corresponding values of \( a \).

1. If \( r = 3 \):
\[ a = \frac{6}{3 - 1} = \frac{6}{2} = 3 \]
The first three terms are:
\[ a = 3, \quad ar = 3 \cdot 3 = 9, \quad ar^2 = 3 \cdot 3^2 = 27 \]

2. If \( r = -3 \):
\[ a = \frac{6}{-3 - 1} = \frac{6}{-4} = -\frac{3}{2} \]
The first three terms are:
\[ a = -\frac{3}{2}, \quad ar = -\frac{3}{2} \cdot (-3) = \frac{9}{2}, \quad ar^2 = -\frac{3}{2} \cdot 9 = -\frac{27}{2} \]

### Step 4: Calculate the sums of the first three terms.

1. For \( r = 3 \):
\[ \text{Sum} = 3 + 9 + 27 = 39 \]

2. For \( r = -3 \):
\[ \text{Sum} = -\frac{3}{2} + \frac{9}{2} - \frac{27}{2} = -\frac{3}{2} + \frac{9 - 27}{2} = -\frac{3}{2} - \frac{18}{2} = -\frac{21}{2} = -10.5 \]

### Final Answer:
The sums of the first three terms of the G.P. are \( 39 \) and \( -10.5 \).

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