https://file.aiquickdraw.com/mathbot/file/1726390020524upjxggei.jpg

The image contains four sequences \( u_n \) that are defined for \( n \geq 0 \), and the task is to determine their monotonicity (whether they are increasing, decreasing, etc.). Here is a breakdown of each sequence:

### 1) Sequence 1:
\[
u_n = n^3 + 9n
\]
We need to determine the monotonicity of this function.

### 2) Sequence 2:
\[
u_n = 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n} \quad (n \geq 0)
\]
This is a harmonic series, and we are tasked with determining whether it increases or decreases.

### 3) Sequence 3:
\[
u_n = \frac{n + 1}{2^n} \quad (n \geq 0)
\]
Here, the behavior of the sequence involves both \( n \) and powers of 2, so we will analyze its monotonicity.

### 4) Sequence 4:
The recurrence relation:
\[
\begin{aligned}
u_0 &= 1 \\
u_{n+1} &= -u_n^2 + u_n - 1 \quad \text{for all } n \geq 0
\end{aligned}
\]
We need to analyze the behavior of this recurrence relation.

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Would you like me to proceed with solving and explaining each of these sequences in detail?

Determine the monotonicity of each of the following sequences: 1) u_n = n^3 + 9n (n >= 0), 2) u_n = 1 + 1/2 + 1/3 + ... + 1/n (n >= 0), 3) u_n = (n + 1) / 2^n (n >= 0), 4) Recurrence relation with u_0 = 1 and u_(n+1) = -u_n^2 + u_n - 1 for all n >= 0.

Analyze the monotonicity of four mathematical sequences, including a cubic polynomial sequence, a harmonic series, an exponential sequence, and a recurrence relation. Understand key mathematical concepts such as sequences, series, monotonicity, and recurrence relations. Suitable for students in Grades 11-12.

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