Find the `8^(th)` term of the sequence :
`(3)/(4),1(1)/(2),3,` . . . . . . . . . . .

Answer

To find the 8th term of the sequence given by \( \frac{3}{4}, \frac{3}{2}, 3, \ldots \), we can follow these steps:

### Step 1: Identify the first term and the common ratio
The first term \( a \) of the sequence is:
\[ a = \frac{3}{4} \]

### Step 2: Determine the second term
The second term is given as \( \frac{3}{2} \).

### Step 3: Calculate the common ratio \( r \)
The common ratio \( r \) can be calculated by dividing the second term by the first term:
\[ r = \frac{\text{second term}}{\text{first term}} = \frac{\frac{3}{2}}{\frac{3}{4}} = \frac{3}{2} \times \frac{4}{3} = \frac{4}{2} = 2 \]

### Step 4: Use the formula for the nth term of a geometric progression
The formula for the \( n \)th term of a geometric progression is given by:
\[ T_n = a \cdot r^{n-1} \]
We need to find the 8th term, so we set \( n = 8 \):
\[ T_8 = a \cdot r^{8-1} = \frac{3}{4} \cdot 2^{7} \]

### Step 5: Calculate \( 2^{7} \)
Calculating \( 2^{7} \):
\[ 2^{7} = 128 \]

### Step 6: Substitute back into the formula
Now we substitute back into the formula:
\[ T_8 = \frac{3}{4} \cdot 128 \]

### Step 7: Simplify the expression
Calculating \( \frac{3 \cdot 128}{4} \):
\[ = \frac{384}{4} = 96 \]

### Final Answer
Thus, the 8th term of the sequence is:
\[ \boxed{96} \]

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