This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Base Case For n=1, we have 9^1 - 1 = 9 - 1 = 8. Since 8 is divisible by 8, the statement is true for n=1. Step 2: Inductive Hypothesis Assume that the statement is true for some positive integer k. This means 9^k - 1 is divisible by 8. So, we can write 9^k - 1 = 8m for some integer m. Rearranging this, we get 9^k = 8m + 1. Step 3: Inductive Step We need to show that the statement is true for n=k+1. Consider 9^k+1 - 1: 9^k+1 - 1 = 9 · 9^k - 1 Substitute 9^k = 8m + 1 from the inductive hypothesis: 9^k+1 - 1 = 9(8m + 1) - 1 9^k+1 - 1 = 72m + 9 - 1 9^k+1 - 1 = 72m + 8 Factor out 8: 9^k+1 - 1 = 8(9m + 1) Since m is an integer, 9m+1 is also an integer. Therefore, 8(9m+1) is divisible by 8. Step 4: Conclusion By the principle of mathematical induction, 9^n - 1 is divisible by 8 for all positive integers n. The statement 9^n - 1 is divisible by 8 for all n N is proven. Send me the next one 📸