Question

Докажите методом математической индукции:

Answer 1

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Answer 2

Proof by Mathematical Induction

To prove a statement using mathematical induction, we typically follow these steps:

1. Base Case: Verify that the statement holds true for the initial value(s) of the variable. 2. Inductive Hypothesis: Assume that the statement is true for some arbitrary value of the variable, denoted as 'k'. 3. Inductive Step: Show that if the statement is true for 'k', it must also be true for 'k+1'.

Let's apply the method of mathematical induction to prove the following statements:

1. $5^n - 4n + 15$ is divisible by 16 for all non-negative integers 'n'. 2. $a_n = 2n^3 + 3n^2 + 7n$ is divisible by 6 for all positive integers 'n'. 3. $4^{2n-1} + 1$ is divisible by 5 for all integers 'n' greater than or equal to 1. 4. $6^{2n-2} + 3^{n+1} + 3^{n-1}$ is divisible by 11 for all integers 'n'.

Statement 1: $5^n - 4n + 15$ is divisible by 16 for all non-negative