Prove by mathematical induction. 1+2+2^2+2^3+...+2^n-1 = 2^n - 1

To prove the given equation by mathematical induction, we need to show that:

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

**Step 1: Base Case (n = 1)**

When \( n = 1 \), the left-hand side of the equation is just the first term of the series:

\[
1 = 2^1 - 1
\]

The right-hand side:

\[
2^1 - 1 = 2 - 1 = 1
\]

So, the base case holds true.

**Step 2: Inductive Hypothesis**

Assume that the equation is true for some integer \( k \geq 1 \). That is, assume:

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

**Step 3: Inductive Step**

We need to show that if the equation holds for \( k \), it also holds for \( k+1 \). That is, we need to prove:

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

Using the inductive hypothesis, the left-hand side up to \( 2^{k-1} \) is equal to \( 2^k - 1 \). Therefore, we can write:

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

Simplifying the right-hand side:

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

Thus, the equation holds for \( k+1 \) as well.

**Step 4: Conclusion**

Since the base case is true, and the inductive step shows that if the equation is true for \( k \), it is also true for \( k+1 \), by the principle of mathematical induction, the equation is true for all positive integers \( n \).

So, we have successfully proven that:

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

Do you want any further details or have any questions?

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Here are 8 related questions to consider:

1. Can you prove the formula for the sum of a geometric series?
2. How does the formula for the sum of this series compare to the sum of an arithmetic series?
3. What are the differences between mathematical induction and strong induction?
4. How would you prove the inequality \( 2^n > n^2 \) for large \( n \) using induction?
5. Can you derive a similar result for summing \( 3^k \) terms?
6. What happens to the sum when \( n \) becomes very large?
7. How can mathematical induction be applied to inequalities?
8. Can you extend this method to prove the formula for the sum of cubes?

**Tip:** Always carefully verify the base case when using mathematical induction, as it's the foundation of your proof.

Learn how to prove the equation 1 + 2 + 2^2 + ... + 2^(n-1) = 2^n - 1 using mathematical induction. This proof covers key concepts including mathematical induction, geometric series, and exponential functions. Suitable for students in Grades 10-12.

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