How can you show that (2^n+3^n) ^1/n is bounded decreasing?

 How can you show that (2^n+3^n) ^1/n is bounded decreasing?👇🏾👇🏾

To show that the sequence $(2^{𝑛}+3^{𝑛}{)}^{1\mathrm{/}𝑛}$ is bounded and decreasing, we need to analyze its behavior as $𝑛$ increases. We will demonstrate both the boundedness and the monotonicity (specifically, that the sequence is decreasing).

Step 1: Boundedness

To establish boundedness, we need to show that $(2^{𝑛}+3^{𝑛}{)}^{1\mathrm{/}𝑛}$ is bounded above and below for all $𝑛\ge 1$.

Upper Bound

For the upper bound, observe that:

$$2^{𝑛}+3^{𝑛}\le 2\cdot 3^{𝑛}=2\cdot 3^{𝑛}$$

Taking the $𝑛$-th root of both sides:

$$(2^{𝑛}+3^{𝑛}{)}^{1\mathrm{/}𝑛}\le (2\cdot 3^{𝑛}{)}^{1\mathrm{/}𝑛}$$

$$(2^{𝑛}+3^{𝑛}{)}^{1\mathrm{/}𝑛}\le 2^{1\mathrm{/}𝑛}\cdot 3$$

As $𝑛\to \mathrm{\infty }$, $2^{1\mathrm{/}𝑛}\to 1$. Therefore, there exists a constant $𝑀$ such that:

$$(2^{𝑛}+3^{𝑛}{)}^{1\mathrm{/}𝑛}\le 3+𝜖$$