Let n in mathbb{N}. Which one of the followin | vedclass

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(D) We test each option for $n \in \mathbb{N}$.$(a)$ For $n=1$, $47^1+16(1)-1 = 47+16-1 = 62$, which is not divisible by $4$.$(b)$ For $n=1$, $2(4^3)-

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$3(5^{2n+1})+2^{3n+1}$ is divisible by $17$

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(D) We test each option for $n \in \mathbb{N}$.$(a)$ For $n=1$, $47^1+16(1)-1 = 47+16-1 = 62$, which is not divisible by $4$.$(b)$ For $n=1$, $2(4^3)-3^4 = 2(64)-81 = 128-81 = 47$, which is not divisible by $9$.$(c)$ For $n=1$, $4^1-3(1)-1 = 4-3-1 = 0$. For $n=2$, $4^2-3(2)-1 = 16-6-1 = 9$, which is not divisible by $11$.$(d)$ Consider $f(n) = 3 \cdot 5^{2n+1} + 2^{3n+1} = 15 \cdot 25^n + 2 \cdot 8^n$.Using the property $x^n - y^n = (x-y)(x^{n-1} + \dots + y^{n-1})$, we can write $25^n = (17+8)^n = 17k + 8^n$.So, $f(n) = 15(17k + 8^n) + 2 \cdot 8^n = 15 \cdot 17k + 15 \cdot 8^n + 2 \cdot 8^n = 15 \cdot 17k + 17 \cdot 8^n = 17(15k + 8^n)$.Thus, $3(5^{2n+1}) + 2^{3n+1}$ is always divisible by $17$ for all $n \in \mathbb{N}$.

Let $n \in \mathbb{N}$. Which one of the following is true?

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