Let P(n) : 3^n

Question

(A) We test the inequality $3^n < n!$ for various values of $n$:For $n = 1: 3^1 = 3, 1! = 1$. $3 < 1$ is False.For $n = 2: 3^2 = 9, 2! = 2$. $9 < 2$ i

Vedclass · Accepted Answer

$7$

Vedclass · Answer

(A) We test the inequality $3^n For $n = 1: 3^1 = 3, 1! = 1$. $3 For $n = 2: 3^2 = 9, 2! = 2$. $9 For $n = 3: 3^3 = 27, 3! = 6$. $27 For $n = 4: 3^4 = 81, 4! = 24$. $81 For $n = 5: 3^5 = 243, 5! = 120$. $243 For $n = 6: 3^6 = 729, 6! = 720$. $729 For $n = 7: 3^7 = 2187, 7! = 5040$. $2187 Since the inequality holds for $n = 7$ and the growth rate of $n!$ is faster than $3^n$ for $n > 6$,the smallest value of $\lambda$ is $7$.

Let $P(n) : 3^n < n!$ for $n \in N$. If this statement is true for all $n \geq \lambda$,then the smallest value of $\lambda$ is:

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