The product of first three terms of a G.P. is `1000`. If we add `6` to its second term and `7` to its third term, the resulting three terms form an A.P. Find the terms of the G.P.

Answer

To solve the problem, we need to find the first three terms of a geometric progression (G.P.) given the conditions in the problem statement. Let's denote the first three terms of the G.P. as \( a \), \( ar \), and \( ar^2 \).

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

1. **Product of the first three terms**:
\[ a \cdot ar \cdot ar^2 = 1000 \]
This simplifies to:
\[ a^3 r^3 = 1000 \]
Taking the cube root gives:
\[ ar = 10 \quad \text{(Equation 1)} \]

2. **Condition for the terms to form an A.P.**:
If we add 6 to the second term and 7 to the third term, the terms become \( a \), \( ar + 6 \), and \( ar^2 + 7 \). For these to be in A.P., the following condition must hold:
\[ (ar + 6) - a = (ar^2 + 7) - (ar + 6) \]
Simplifying this gives:
\[ ar + 6 - a = ar^2 + 7 - ar - 6 \]
Rearranging leads to:
\[ ar + 6 - a = ar^2 + 1 - ar \]
This simplifies to:
\[ ar + 6 - a = ar^2 - ar + 1 \]
Rearranging gives:
\[ ar + 6 - a = ar^2 - ar + 1 \]
Thus:
\[ ar + 6 - a = ar^2 - ar + 1 \]
Rearranging gives:
\[ ar^2 - 2ar + a - 5 = 0 \quad \text{(Equation 2)} \]

### Step 2: Substitute \( ar = 10 \) into Equation 2.

From Equation 1, we know \( ar = 10 \). Substituting this into Equation 2:
\[ ar^2 - 2(10) + a - 5 = 0 \]
This simplifies to:
\[ ar^2 - 20 + a - 5 = 0 \]
Thus:
\[ ar^2 + a - 25 = 0 \quad \text{(Equation 3)} \]

### Step 3: Express \( a \) in terms of \( r \).

From Equation 1, we have:
\[ a = \frac{10}{r} \]
Substituting this into Equation 3 gives:
\[ 10r + \frac{10}{r} - 25 = 0 \]
Multiplying through by \( r \) to eliminate the fraction:
\[ 10r^2 + 10 - 25r = 0 \]
This simplifies to:
\[ 10r^2 - 25r + 10 = 0 \]
Dividing through by 5 gives:
\[ 2r^2 - 5r + 2 = 0 \]

### Step 4: Solve the quadratic equation.

Using the quadratic formula \( r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ r = \frac{5 \pm \sqrt{(-5)^2 - 4 \cdot 2 \cdot 2}}{2 \cdot 2} \]
Calculating the discriminant:
\[ r = \frac{5 \pm \sqrt{25 - 16}}{4} \]
This simplifies to:
\[ r = \frac{5 \pm 3}{4} \]
Thus, we have two possible values for \( r \):
1. \( r = 2 \)
2. \( r = \frac{1}{2} \)

### Step 5: Find corresponding values of \( a \).

1. **For \( r = 2 \)**:
\[ a = \frac{10}{2} = 5 \]
The terms are:
\[ 5, 10, 20 \]

2. **For \( r = \frac{1}{2} \)**:
\[ a = \frac{10}{\frac{1}{2}} = 20 \]
The terms are:
\[ 20, 10, 5 \]

### Final Answer:
The terms of the G.P. are \( 5, 10, 20 \) or \( 20, 10, 5 \).

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