https://file.aiquickdraw.com/mathbot/file/1727859974298xkz4dxwb.jpg

The given problem asks us to prove by **mathematical induction** that the expression \(3^n + 7^n - 2\) is divisible by 8 for \(n \geq 1\). Let's proceed step by step.

### Step 1: Base Case (n = 1)
We need to check if the statement is true for \(n = 1\).

\[
3^1 + 7^1 - 2 = 3 + 7 - 2 = 8
\]

Since 8 is divisible by 8, the base case holds true.

### Step 2: Inductive Hypothesis
Assume the statement is true for some \(n = k\), where \(k \geq 1\). That is:

\[
3^k + 7^k - 2 = 8m, \quad \text{for some integer } m.
\]

This means that \(3^k + 7^k - 2\) is divisible by 8.

### Step 3: Inductive Step (n = k + 1)
We need to prove that the statement holds for \(n = k + 1\). That is, we need to show that:

\[
3^{k+1} + 7^{k+1} - 2
\]

is divisible by 8. Let's express \(3^{k+1}\) and \(7^{k+1}\) in terms of \(3^k\) and \(7^k\):

\[
3^{k+1} = 3 \cdot 3^k, \quad 7^{k+1} = 7 \cdot 7^k
\]

Now consider the expression:

\[
3^{k+1} + 7^{k+1} - 2 = 3 \cdot 3^k + 7 \cdot 7^k - 2
\]

Using the inductive hypothesis, we know that:

\[
3^k + 7^k - 2 = 8m \quad \text{(for some integer } m\text{)}
\]

To proceed further, let's look at the differences between \(3^{k+1} + 7^{k+1} - 2\) and \(3^k + 7^k - 2\):

\[
3^{k+1} + 7^{k+1} - 2 - (3^k + 7^k - 2)
\] 

This

Use mathematical induction to prove that (3^n + 7^n - 2) is divisible by 8 for n ≥ 1.

Learn how to prove that (3^n + 7^n - 2) is divisible by 8 using mathematical induction. This step-by-step solution includes the base case, inductive hypothesis, and inductive step. Suitable for Grade 11-12 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation