The greatest integer less than or equal to the sum of first 100 terms of the sequence `1/3, 5/9,19/27,65/81 ,........` is equal to ______

Answer

To find the greatest integer less than or equal to the sum of the first 100 terms of the sequence \( \frac{1}{3}, \frac{5}{9}, \frac{19}{27}, \frac{65}{81}, \ldots \), we can start by identifying a pattern in the terms of the sequence.

### Step 1: Identify the general term of the sequence

The given terms are:
- \( a_1 = \frac{1}{3} = \frac{3^1 - 2^1}{3^1} \)
- \( a_2 = \frac{5}{9} = \frac{3^2 - 2^2}{3^2} \)
- \( a_3 = \frac{19}{27} = \frac{3^3 - 2^3}{3^3} \)
- \( a_4 = \frac{65}{81} = \frac{3^4 - 2^4}{3^4} \)

From this, we can see that the \( n \)-th term can be expressed as:
\[ a_n = \frac{3^n - 2^n}{3^n} \]

### Step 2: Write the sum of the first 100 terms

The sum of the first 100 terms \( S_{100} \) can be expressed as:
\[ S_{100} = \sum_{n=1}^{100} a_n = \sum_{n=1}^{100} \left( 1 - \frac{2^n}{3^n} \right) \]
This simplifies to:
\[ S_{100} = 100 - \sum_{n=1}^{100} \frac{2^n}{3^n} \]

### Step 3: Calculate the sum of the geometric series

The series \( \sum_{n=1}^{100} \frac{2^n}{3^n} \) is a geometric series with:
- First term \( a = \frac{2}{3} \)
- Common ratio \( r = \frac{2}{3} \)
- Number of terms \( n = 100 \)

The sum of the first \( n \) terms of a geometric series is given by:
\[ S_n = a \frac{1 - r^n}{1 - r} \]
Thus, we have:
\[ \sum_{n=1}^{100} \frac{2^n}{3^n} = \frac{2/3 \cdot (1 - (2/3)^{100})}{1 - 2/3} = 2 \cdot (1 - (2/3)^{100}) \]

### Step 4: Substitute back into the sum

Now substituting this back into our expression for \( S_{100} \):
\[ S_{100} = 100 - 2 \cdot (1 - (2/3)^{100}) = 100 - 2 + 2 \cdot (2/3)^{100} \]
\[ S_{100} = 98 + 2 \cdot (2/3)^{100} \]

### Step 5: Estimate the value of \( (2/3)^{100} \)

Since \( \left( \frac{2}{3} \right)^{100} \) is a very small number (as \( \frac{2}{3} < 1 \)), we can approximate:
\[ (2/3)^{100} \approx 0 \]
Thus:
\[ S_{100} \approx 98 + 0 = 98 \]

### Step 6: Find the greatest integer less than or equal to \( S_{100} \)

The greatest integer less than or equal to \( S_{100} \) is:
\[ \lfloor S_{100} \rfloor = 98 \]

### Final Answer
The greatest integer less than or equal to the sum of the first 100 terms of the sequence is \( \boxed{98} \).