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This question involves finding the sum of a geometric series. Let's go step-by-step to understand the solution.

### Problem Breakdown
We are given the series:
\[
1 + 2 + 2^2 + 2^3 + \dots + 2^{64}
\]
This is a **geometric series** where:
- The first term \( a = 1 \)
- The common ratio \( r = 2 \)
- The number of terms \( n = 65 \) (since it goes up to \( 2^{64} \))

### Step-by-Step Solution

1. **Formula for the Sum of a Geometric Series**:
   The sum \( S_n \) of the first \( n \) terms of a geometric series with common ratio \( r \) is:
   \[
   S_n = a \frac{r^n - 1}{r - 1}
   \]
   
2. **Substitute the Known Values**:
   In this case:
   - \( a = 1 \)
   - \( r = 2 \)
   - \( n = 65 \) (since the last term is \( 2^{64} \))

   Therefore:
   \[
   S_{65} = \frac{2^{65} - 1}{2 - 1} = 2^{65} - 1
   \]

3. **Simplify the Expression**:
   \[
   2^{65} - 1
   \]

4. **Convert to Scientific Notation**:
   The expression \( 2^{65} \) is a very large number. Using a calculator or approximation:
   \[
   2^{65} \approx 3.69 \times 10^{19}
   \]
   
   Therefore:
   \[
   2^{65} - 1 \approx 3.69 \times 10^{19} \, (\text{rounded to 3 significant figures})
   \]

### Final Answer
The sum of the series is approximately \( 3.69 \times 10^{19} \).

Would you like more details or have any questions?

### Related Questions
1. How can you derive the formula for the sum of a geometric series?
2. What would be the sum if the series were to continue up to \( 2^{100} \)?
3. How does changing the common ratio affect the sum of the geometric series?
4. What is the difference between a geometric series and an arithmetic series?
5. Can the formula for the sum of an infinite geometric series apply here?

### Tip:
For geometric series, if the common ratio \( r \) is greater than 1, the series grows rapidly, and the sum formula helps manage large numbers efficiently.

Learn how to calculate the sum of a geometric series 1 + 2 + 2^2 + ... + 2^64 using the geometric series sum formula. This problem covers key concepts such as exponents, series, and scientific notation. Suitable for students in Grades 9-12.

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