Use Mathematical Induction to show that 1+2+4+⋯=2𝑛+1−1.

To prove the statement \( 1 + 2 + 4 + \cdots + 2^n = 2^{n+1} - 1 \) using mathematical induction, we will follow these steps:

### Step 1: **Base Case**

First, we need to verify the statement for the base case \( n = 0 \).

For \( n = 0 \), the left-hand side of the equation is just the first term in the series:

\[
1 = 2^0
\]

The right-hand side is:

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

Since both sides are equal, the base case holds.

### Step 2: **Inductive Hypothesis**

Next, assume that the statement is true for some arbitrary positive integer \( k \). That is, assume:

\[
1 + 2 + 4 + \cdots + 2^k = 2^{k+1} - 1
\]

This assumption is called the **inductive hypothesis**.

### Step 3: **Inductive Step**

Now, we need to prove that the statement is true for \( n = k + 1 \). That is, we need to show:

\[
1 + 2 + 4 + \cdots + 2^k + 2^{k+1} = 2^{(k+1) + 1} - 1
\]

Starting from the inductive hypothesis, add \( 2^{k+1} \) to both sides:

\[
1 + 2 + 4 + \cdots + 2^k + 2^{k+1} = \left(2^{k+1} - 1\right) + 2^{k+1}
\]

Simplify the right-hand side:

\[
1 + 2 + 4 + \cdots + 2^k + 2^{k+1} = 2^{k+1} - 1 + 2^{k+1} = 2 \cdot 2^{k+1} - 1 = 2^{k+2} - 1
\]

This is exactly the form we need for \( n = k + 1 \), which is:

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

### Step 4: **Conclusion**

Since the base case holds, and the inductive step has been proven, by mathematical induction, the statement

\[
1 + 2 + 4 + \cdots + 2^n = 2^{n+1} - 1
\]

is true for all non-negative integers \( n \).

Would you like more details on any specific part, or do you have any questions?

---

Here are some related questions to consider:

1. What are the other common examples where mathematical induction is used?
2. How can mathematical induction be applied to inequalities?
3. What is the difference between weak and strong induction?
4. How does the sum of a geometric series relate to this problem?
5. Can you extend this formula to other types of series?

**Tip:** When using mathematical induction, always ensure that the base case is clearly established; it is the foundation upon which the entire proof is built.

Learn how to use mathematical induction to prove the sum of a geometric series 1 + 2 + 4 + ⋯ = 2𝑛 + 1 − 1. This problem demonstrates the application of mathematical induction, covering geometric series and exponential growth. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation