1+2+2²+...2n-1=2n-1 solve this using mathematical induction

To prove the statement \( 1 + 2 + 2^2 + \dots + 2^{n-1} = 2^n - 1 \) by mathematical induction, we proceed as follows:

### Step 1: Base Case
We check the statement for \( n = 1 \).

- **Left-hand side (LHS):** For \( n = 1 \), the sum is \( 1 \) (since the series only has the first term, \( 2^{1-1} = 2^0 = 1 \)).
- **Right-hand side (RHS):** \( 2^1 - 1 = 2 - 1 = 1 \).

Since LHS = RHS, the base case holds true.

### Step 2: Inductive Hypothesis
Assume that the statement is true for some arbitrary positive integer \( k \). That is, assume:

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

### Step 3: Inductive Step
We need to show that the statement is true for \( k+1 \), i.e., we want to prove:

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

Start with the left-hand side of the equation for \( n = k + 1 \):

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

By the inductive hypothesis, we know:

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

So, we can substitute this into the equation:

\[
2^k - 1 + 2^k
\]

Simplify the expression:

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

This is exactly the right-hand side of the equation for \( n = k + 1 \). Thus, the inductive step holds.

### Conclusion:
Since the base case holds, and the inductive step has been proven, by the principle of mathematical induction, the statement \( 1 + 2 + 2^2 + \dots + 2^{n-1} = 2^n - 1 \) is true for all positive integers \( n \).

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Would you like more details on any part of this proof, or do you have any questions? Here are some related questions you might find interesting:

1. How does mathematical induction differ from direct proof?
2. Can this approach be used to prove inequalities?
3. How does this formula relate to the sum of a geometric series?
4. Can you prove similar formulas using mathematical induction?
5. What are the limitations of using mathematical induction?

**Tip:** Always verify the base case carefully, as the entire inductive proof depends on its correctness.

Learn how to prove the sum of a geometric series using mathematical induction. This detailed solution covers the base case, inductive hypothesis, and inductive step, demonstrating the application of the principle of mathematical induction. Suitable for high school students in Grades 10-12.

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