$$ {\displaylines\sum2^{k}\\ \\ \placeholder{}\\ } $$

It seems like you're working on summing powers of 2, but the expression is incomplete. To better assist you, I'll need to know the summation limits and variable you're summing over.

Typically, summations of powers of 2 look like this:

\[
\sum_{k=0}^{n} 2^k
\]

This is a geometric series with a common ratio of 2, and its closed form can be computed using the formula for the sum of a geometric series:

\[
\sum_{k=0}^{n} 2^k = 2^{n+1} - 1
\]

If you'd like to complete the expression or clarify the range of summation, I'd be happy to provide further details!

### Would you like to:
- Specify the summation limits or bounds for $ k $?
- Expand or simplify other parts of this expression?
- Discuss a specific property of geometric series?

#### Related Questions:
1. How do you derive the formula for the sum of a geometric series?
2. What happens to the sum when the common ratio is different from 2?
3. How can the formula for infinite geometric series be applied?
4. Can this sum be used to solve binary-related problems?
5. What is the practical application of summing powers of 2 in computing?

**Tip**: Geometric series with powers of 2 often arise in computing contexts, particularly in binary arithmetic and algorithms.

$$ \sum_{k=0}^{n} 2^k \ \ \placeholder{} $$

Learn how to sum powers of 2 using the geometric series formula. This solution walks through the key concepts and formulas involved in calculating the sum of powers of 2, such as $ \sum_{k=0}^{n} 2^k = 2^{n+1} - 1 $. Suitable for students in Grades 9-12.

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