Find the indicated term of the following G.P.:
`12, 8, 16/3, …………..t_(10)`

Answer

To find the 10th term of the given geometric progression (G.P.) \(12, 8, \frac{16}{3}, \ldots\), we can follow these steps:

### Step 1: Identify the first term (A)
The first term \(A\) of the G.P. is:
\[ A = 12 \]

**Hint:** The first term is simply the first number in the sequence.

### Step 2: Calculate the common ratio (R)
The common ratio \(R\) can be found by dividing the second term by the first term:
\[ R = \frac{\text{second term}}{\text{first term}} = \frac{8}{12} = \frac{2}{3} \]

**Hint:** The common ratio is found by dividing any term by the previous term.

### Step 3: Determine the term number (n)
We need to find the 10th term, so:
\[ n = 10 \]

**Hint:** The term number is given in the problem statement.

### Step 4: Use the formula for the nth term of a G.P.
The formula for the nth term \(T_n\) of a G.P. is given by:
\[ T_n = A \cdot R^{n-1} \]
Substituting the values we have:
\[ T_{10} = 12 \cdot \left(\frac{2}{3}\right)^{10-1} \]
\[ T_{10} = 12 \cdot \left(\frac{2}{3}\right)^{9} \]

**Hint:** Remember to subtract 1 from the term number when using the formula.

### Step 5: Calculate \(\left(\frac{2}{3}\right)^{9}\)
Now we need to calculate \(\left(\frac{2}{3}\right)^{9}\):
\[ \left(\frac{2}{3}\right)^{9} = \frac{2^9}{3^9} = \frac{512}{19683} \]

**Hint:** Use the properties of exponents to simplify the calculation.

### Step 6: Multiply by the first term
Now we can find \(T_{10}\):
\[ T_{10} = 12 \cdot \frac{512}{19683} = \frac{6144}{19683} \]

**Hint:** When multiplying fractions, multiply the numerators and the denominators separately.

### Final Answer
Thus, the 10th term \(T_{10}\) of the G.P. is:
\[ T_{10} = \frac{6144}{19683} \]