The given expression is a sum of powers of 3, which can be written as:

$S = 3^{100} + 3^{99} + 3^{98} + \cdots + 3^5$

This is a finite geometric series, where the first term $a = 3^5$, the common ratio $r = 3$, and the number of terms is $100 - 5 + 1 = 96$.

The sum of a finite geometric series is given by the formula:

$S_n = a \cdot \frac{r^n - 1}{r - 1}$

where:

• $a$ is the first term,
• $r$ is the common ratio,
• $n$ is the number of terms.

Substitute the values:

• $a = 3^5$,
• $r = 3$,
• $n = 96$.

Thus, the sum becomes:

$S = 3^5 \cdot \frac{3^{96} - 1}{3 - 1}$ $S = 3^5 \cdot \frac{3^{96} - 1}{2}$

Now, calculate the value of $3^5$:

$3^5 = 243$

So, the sum is:

$S = 243 \cdot \frac{3^{96} - 1}{2}$

This is the simplified form of the sum. It involves a large power of 3, but the expression gives the general result. To compute it exactly, you would typically need a calculator or software to handle the large powers.

Let me know if you need more details on this or have any other questions!

Here are some related questions:

1. What is the sum of the first 100 terms of the geometric series $3^0 + 3^1 + 3^2 + \cdots$?
2. How do we handle the sum of a geometric series with negative common ratio?
3. How can the geometric series formula be derived?
4. What happens if we subtract the sum of a geometric series from its successive term?
5. How do you apply this geometric series sum to model exponential growth?

Tip: When dealing with large powers in geometric sums, using logarithms or software tools like Python or Wolfram Alpha can help with computation!