Answer:
To prove that 5^n + 2 (11^n) is divisible by 3 for any integer n ≥ 1, we can use mathematical induction.
Base Step: For n = 1, 5^1 + 2 (11^1) = 5 + 22 = 27, which is divisible by 3.
Inductive Step: Assume that the statement is true for some k ≥ 1, i.e., 5^k + 2 (11^k) is divisible by 3. We need to show that the statement is also true for k+1, i.e., 5^(k+1) + 2 (11^(k+1)) is divisible by 3.
We have:
5^(k+1) + 2 (11^(k+1)) = 5^k * 5 + 2 * 11 * 11^k = 5^k * 5 + 2 * 3 * 3 * 11^k = 5^k * 5 + 6 * 3^2 * 11^k
Now, we notice that 5^k * 5 is divisible by 3 (because 5 is not divisible by 3, and therefore 5^k is not divisible by 3, which means that 5^k * 5 is divisible by 3). Also, 6 * 3^2 * 11^k is clearly divisible by 3.
Therefore, we can conclude that 5^(k+1) + 2 (11^(k+1)) is divisible by 3.
By mathematical induction, we have proved that for any integer n ≥ 1, 5^n + 2 (11^n) is divisible by 3
Step-by-step explanation: