Find the next three terms of the sequence : 36,12,4, . . . . . . . . . . . .

Answer

To find the next three terms of the sequence: 36, 12, 4, we will first determine if this sequence is a geometric progression (GP) and then calculate the next terms.

### Step 1: Identify the first three terms
Let:
- \( t_1 = 36 \)
- \( t_2 = 12 \)
- \( t_3 = 4 \)

### Step 2: Check if the sequence is a GP
To check if the sequence is a geometric progression, we need to see if the ratio of consecutive terms is constant. We calculate the ratios:

\[ \text{Ratio } r = \frac{t_2}{t_1} = \frac{12}{36} = \frac{1}{3} \]

\[ \text{Ratio } r = \frac{t_3}{t_2} = \frac{4}{12} = \frac{1}{3} \]

Since both ratios are equal, the sequence is indeed a geometric progression with a common ratio \( r = \frac{1}{3} \).

### Step 3: Find the next term \( t_4 \)
The formula for the \( n \)-th term of a geometric progression is given by:

\[ t_n = a \cdot r^{(n-1)} \]

Where:
- \( a \) is the first term (36)
- \( r \) is the common ratio \( \left(\frac{1}{3}\right) \)
- \( n \) is the term number

To find \( t_4 \):

\[ t_4 = 36 \cdot \left(\frac{1}{3}\right)^{3} = 36 \cdot \frac{1}{27} = \frac{36}{27} = \frac{4}{3} \]

### Step 4: Find the next term \( t_5 \)
Now we calculate \( t_5 \):

\[ t_5 = 36 \cdot \left(\frac{1}{3}\right)^{4} = 36 \cdot \frac{1}{81} = \frac{36}{81} = \frac{4}{9} \]

### Step 5: Find the next term \( t_6 \)
Finally, we calculate \( t_6 \):

\[ t_6 = 36 \cdot \left(\frac{1}{3}\right)^{5} = 36 \cdot \frac{1}{243} = \frac{36}{243} = \frac{4}{27} \]

### Conclusion
The next three terms of the sequence are:
- \( t_4 = \frac{4}{3} \)
- \( t_5 = \frac{4}{9} \)
- \( t_6 = \frac{4}{27} \)