In a G.P., `T_(10) = 9 and T_(4) = 4`, then `T_(7) =`

Options

A. `4//9`
B. `9//4`
C. 6
D. 36

Answer

To solve the problem step by step, we start with the information given in the question.

### Step 1: Define the terms of the G.P.
Let the first term of the G.P. be \( A \) and the common ratio be \( r \).

### Step 2: Write the equations for \( T_{10} \) and \( T_{4} \)
From the properties of a G.P., we know that:
- \( T_{10} = A r^{9} = 9 \) (Equation 1)
- \( T_{4} = A r^{3} = 4 \) (Equation 2)

### Step 3: Divide Equation 1 by Equation 2
To eliminate \( A \), we can divide Equation 1 by Equation 2:
\[ \frac{A r^{9}}{A r^{3}} = \frac{9}{4} \]
This simplifies to:
\[ r^{6} = \frac{9}{4} \]

### Step 4: Solve for \( r^{6} \)
Thus, we have:
\[ r^{6} = \frac{9}{4} \]

### Step 5: Find \( T_{7} \)
Now, we need to find \( T_{7} \):
\[ T_{7} = A r^{6} \]
Substituting the value of \( r^{6} \) from Step 4:
\[ T_{7} = A \cdot \frac{9}{4} \]

### Step 6: Find \( A \)
To find \( A \), we can use Equation 2:
\[ A r^{3} = 4 \]
We can express \( A \) in terms of \( r^{3} \):
\[ A = \frac{4}{r^{3}} \]

### Step 7: Substitute \( A \) back into \( T_{7} \)
Now substituting \( A \) into the expression for \( T_{7} \):
\[ T_{7} = \frac{4}{r^{3}} \cdot \frac{9}{4} \]
This simplifies to:
\[ T_{7} = \frac{9}{r^{3}} \]

### Step 8: Find \( r^{3} \)
To find \( r^{3} \), we can use the relation \( r^{6} = \frac{9}{4} \):
\[ r^{3} = \sqrt{r^{6}} = \sqrt{\frac{9}{4}} = \frac{3}{2} \]

### Step 9: Substitute \( r^{3} \) back into \( T_{7} \)
Now substituting \( r^{3} \) into the expression for \( T_{7} \):
\[ T_{7} = \frac{9}{\frac{3}{2}} = 9 \cdot \frac{2}{3} = 6 \]

### Final Answer
Thus, \( T_{7} = 6 \).

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