Final answer:
By applying the principle of mathematical induction, it is proven that 21 divides 4^(n+1) + 5^(2n−1) for every positive integer n.
Step-by-step explanation:
To prove that 21 divides 4(n+1) + 5(2n−1) for all positive integers n, we will use the concept of mathematical induction. The base case for n=1 can be easily verified: 42 + 5 = 21, which is divisible by 21.
Assuming the statement is true for a particular k, meaning 4(k+1) + 5(2k−1) is divisible by 21, we must show it holds for k+1. The nth term can be expressed as 4(k+2) + 5(2(k+1)−1) = 4 × 4(k+1) + 25 × 5(2k−1).
Given that 21 divides both 43 and 52, and both of these terms appear in the expansion, it follows that the expression is divisible by 21 for any n. This concludes the proof of the divisibility by 21 using mathematical induction.