To prove that 7n - 6n - 1 is divisible by 36 for all n ∈ N, we can use the principle of mathematical induction.

Step 1: Base Case
Let's start with n = 1.
7^1 - 6^1 - 1 = 0, which is divisible by 36.

Step 2: Inductive Step
Assume that the statement is true for some n = k, i.e., 7^k - 6^k - 1 is divisible by 36.

Step 3: Inductive Hypothesis
We need to prove that the statement is true for n = k + 1, i.e., 7^(k+1) - 6^(k+1) - 1 is divisible by 36.

Let's expand 7^(k+1) - 6^(k+1) - 1:
= 7*7^k - 6*6^k - 1
= 7*(7^k - 6^k - 1) + 42^k - 6^k
= 7*(7^k - 6^k - 1) + 36^k + 6^k - 6^k
= 7*(7^k - 6^k - 1) + 36^k

By the inductive hypothesis, 7^k - 6^k - 1 is divisible by 36, so 7*(7^k - 6^k - 1) is also divisible by 36. And 36^k is obviously divisible by 36.

Therefore, 7^(k+1) - 6^(k+1) - 1 is divisible by 36, which completes the proof by induction.

Question

To prove that 7n - 6n - 1 is divisible by 36 for all n ∈ N, we can use the principle of mathematical induction.

Step 1: Base Case
Let's start with n = 1. 
7^1 - 6^1 - 1 = 0, which is divisible by 36.

Step 2: Inductive Step
Assume that the statement is true for some n = k, i.e., 7^k - 6^k - 1 is divisible by 36.

Step 3: Inductive Hypothesis
We need to prove that the statement is true for n = k + 1, i.e., 7^(k+1) - 6^(k+1) - 1 is divisible by 36.

Let's expand 7^(k+1) - 6^(k+1) - 1:
= 7*7^k - 6*6^k - 1
= 7*(7^k - 6^k - 1) + 42^k - 6^k
= 7*(7^k - 6^k - 1) + 36^k + 6^k - 6^k
= 7*(7^k - 6^k - 1) + 36^k

By the inductive hypothesis, 7^k - 6^k - 1 is divisible by 36, so 7*(7^k - 6^k - 1) is also divisible by 36. And 36^k is obviously divisible by 36.

Therefore, 7^(k+1) - 6^(k+1) - 1 is divisible by 36, which completes the proof by induction.

Knowee AI · Accepted Answer

To prove that 7n - 6n - 1 is divisible by 36 for all n ∈ N, we can use the principle of mathematical induction.

Step 1: Base Case
Let's start with n = 1. 
7^1 - 6^1 - 1 = 0, which is divisible by 36.

Step 2: Inductive Step
Assume that the statement is true for some n = k, i.e., 7^k - 6^k - 1 is divisible by 36.

Step 3: Inductive Hypothesis
We need to prove that the statement is true for n = k + 1, i.e., 7^(k+1) - 6^(k+1) - 1 is divisible by 36.

Let's expand 7^(k+1) - 6^(k+1) - 1:
= 7*7^k - 6*6^k - 1
= 7*(7^k - 6^k - 1) + 42^k - 6^k
= 7*(7^k - 6^k - 1) + 36^k + 6^k - 6^k
= 7*(7^k - 6^k - 1) + 36^k

By the inductive hypothesis, 7^k - 6^k - 1 is divisible by 36, so 7*(7^k - 6^k - 1) is also divisible by 36. And 36^k is obviously divisible by 36.

Therefore, 7^(k+1) - 6^(k+1) - 1 is divisible by 36, which completes the proof by induction.

Problem 3. Prove that 7n − 6n − 1 is divisible by 36 for all n ∈ N.

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