Hence show that 5*7^n+3*11^n-8 is divisible by 60 for all natural numbers n
To prove that 5 * 7^n + 3 * 11^n - 8 is divisible by 60 for all natural numbers n, we can use mathematical induction.
Step 1: Base Case
First, we need to prove that the given expression is divisible by 60 when n = 1.
Substituting n = 1 in the expression, we get:
5 * 7^1 + 3 * 11^1 - 8 = 5 * 7 + 3 * 11 - 8 = 35 + 33 - 8 = 60
Since the expression is equal to 60 when n = 1, it is divisible by 60.
Step 2: Inductive Hypothesis
Assume that the expression 5 * 7^k + 3 * 11^k - 8 is divisible by 60 for some arbitrary positive integer k.
Step 3: Inductive Step
We need to prove that if the expression is divisible by 60 for k, it is also divisible by 60 for k + 1.
Substituting n = k + 1 in the expression, we get:
5 * 7^(k+1) + 3 * 11^(k+1) - 8 = 5 * 7 * 7^k + 3 * 11 * 11^k - 8 = 35 * 7^k + 33 * 11^k + 13 * 7^k - 8
Now, we can rewrite this expression as:
35 * 7^k + 13 * 7^k + 33 * 11^k - 8
We know that 35 * 7^k + 13 * 7^k is divisible by 60 (from the inductive hypothesis). Also, 33 * 11^k is divisible by 60 because 33 is divisible by 3 and 11^k is divisible by 1 (since 11^k is a positive integer). Finally, -8 is divisible by 60 because -8/60 is an integer.
Therefore, the entire expression is divisible by 60.
Step 4: Conclusion
By mathematical induction, we have shown that for all natural numbers n, the expression 5 * 7^n + 3 * 11^n - 8 is divisible by 60.