Final answer:
To prove that 8n-1 is divisible by 7 for all integers n≥1, we can use mathematical induction. By verifying the base case and the inductive case, we can conclude that the statement holds true for all values of n≥1.
Step-by-step explanation:
To prove that 8n-1 is divisible by 7 for all integers n≥1, let's use mathematical induction.
Step 1: Base Case: Substitute n=1 into the expression 8n-1 and check if it is divisible by 7. 8(1)-1 = 7, which is divisible by 7.
Step 2: Inductive Case: Assume that 8n-1 is divisible by 7 for a certain value of n=k.
Step 3: Prove for n=k+1: We need to show that 8(k+1)-1 is divisible by 7. We can rewrite this expression as 8k + 7 = 7k + k + 7. We already assumed that 8k-1 is divisible by 7, and we know that 7k + k + 7 is also divisible by 7.
Therefore, by mathematical induction, 8n-1 is divisible by 7 for all integers n≥1.