The values of the natural numbers n for which | vedclass

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(A) We test the inequality $2^n > 2n + 1$ for natural numbers $n = 1, 2, 3, 4, \dots$ For $n = 1$: $2^1 = 2$ and $2(1) + 1 = 3$. Since $2 > 3$ is fals

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For $n \ge 3$

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(A) We test the inequality $2^n > 2n + 1$ for natural numbers $n = 1, 2, 3, 4, \dots$ For $n = 1$: $2^1 = 2$ and $2(1) + 1 = 3$. Since $2 > 3$ is false,the inequality does not hold for $n = 1$. For $n = 2$: $2^2 = 4$ and $2(2) + 1 = 5$. Since $4 > 5$ is false,the inequality does not hold for $n = 2$. For $n = 3$: $2^3 = 8$ and $2(3) + 1 = 7$. Since $8 > 7$ is true,the inequality holds for $n = 3$. For $n = 4$: $2^4 = 16$ and $2(4) + 1 = 9$. Since $16 > 9$ is true,the inequality holds for $n = 4$. By mathematical induction,it can be shown that the inequality holds for all $n \ge 3$.

The values of the natural numbers $n$ for which the inequality $2^n > 2n + 1$ is valid are:

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