6) Use Mathematical Induction to show that 1+2+2 2 + 2 3 + ⋯ … … … + 2 𝑛 = 2 𝑛+1 − 1

To prove the statement

\[
1 + 2 + 2^2 + 2^3 + \dots + 2^n = 2^{n+1} - 1
\]

using **mathematical induction**, we proceed in two steps:

### Step 1: Base Case
We start by verifying the formula for \( n = 1 \).

For \( n = 1 \):
\[
1 + 2 = 2^{1+1} - 1 = 2^2 - 1 = 4 - 1 = 3
\]

Thus, the base case holds since \( 1 + 2 = 3 \).

### Step 2: Inductive Step
Assume that the statement is true for some \( n = k \); that is, we assume

\[
1 + 2 + 2^2 + 2^3 + \dots + 2^k = 2^{k+1} - 1
\]

This is our **inductive hypothesis**. We need to show that this assumption implies the statement is true for \( n = k+1 \), i.e.,

\[
1 + 2 + 2^2 + 2^3 + \dots + 2^k + 2^{k+1} = 2^{(k+1)+1} - 1 = 2^{k+2} - 1
\]

Starting from the left-hand side, we add \( 2^{k+1} \) to both sides of the inductive hypothesis:

\[
1 + 2 + 2^2 + 2^3 + \dots + 2^k + 2^{k+1} = (2^{k+1} - 1) + 2^{k+1}
\]

Now, simplify the right-hand side:

\[
(2^{k+1} - 1) + 2^{k+1} = 2^{k+1} + 2^{k+1} - 1 = 2 \cdot 2^{k+1} - 1 = 2^{k+2} - 1
\]

Thus, we have shown that if the statement holds for \( n = k \), it also holds for \( n = k+1 \).

### Conclusion
By mathematical induction, the formula is true for all \( n \geq 1 \):

\[
1 + 2 + 2^2 + 2^3 + \dots + 2^n = 2^{n+1} - 1
\]

Would you like additional details on any part of the solution? 

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### Related Questions
1. How does mathematical induction work for other series?
2. Can this formula be derived using a different approach?
3. How does induction apply to inequalities instead of equalities?
4. What other types of mathematical proofs complement induction?
5. What are some common applications of summing powers of 2?

**Tip:** In mathematical induction, always check that the base case holds; it ensures that the pattern starts correctly.

Use Mathematical Induction to show that 1 + 2 + 2^2 + 2^3 + ⋯ + 2^n = 2^(n+1) - 1

Discover how to use mathematical induction to prove that the sum of powers of 2, from 1 up to 2^n, equals 2^(n+1) - 1. This step-by-step solution includes key concepts like induction, series, and powers of 2. Perfect for high school students in Grades 10-12.

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