Find the `9^(th)` term of the series :
1,4,16,64 . . . . . . . . . . .

Answer

To find the 9th term of the series: 1, 4, 16, 64, ..., we can follow these steps:

### Step 1: Identify the first term (a) and the common ratio (r)
The series given is: 1, 4, 16, 64.

- The first term \( a \) is 1.
- To find the common ratio \( r \), we can divide the second term by the first term:
\[ r = \frac{\text{Second term}}{\text{First term}} = \frac{4}{1} = 4 \]

### Step 2: Use the formula for the nth term of a geometric progression
The formula for the nth term \( T_n \) of a geometric progression is given by:
\[ T_n = a \cdot r^{(n-1)} \]
where:
- \( a \) is the first term,
- \( r \) is the common ratio,
- \( n \) is the term number.

### Step 3: Substitute the values into the formula
We want to find the 9th term, so we set \( n = 9 \):
\[ T_9 = 1 \cdot 4^{(9-1)} = 1 \cdot 4^8 \]

### Step 4: Calculate \( 4^8 \)
Now we need to calculate \( 4^8 \):
\[ 4^8 = (2^2)^8 = 2^{16} = 65536 \]

### Step 5: Conclusion
Thus, the 9th term of the series is:
\[ T_9 = 65536 \]

### Final Answer
The 9th term of the series is **65536**.

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