$$ 17\text{. (}15分)\text{ 已知函数}f\left(x\right)=\sin^2x+\sin x+\cos^2x\text{.} $$
$$ \text{(}1)\left(6分\right)求函数f\left(x\right)的对称中心\text{.} $$
$$ \left(2\right)\,\left(9分\right)\,若函数f\left(x\right)在区间\left\lbrack0,\text{ }\frac{n\pi}{2}\right\rbrack上恰好有2024个零点\text{, 求实数}n的取值范围\text{.} $$

Let's solve the problem step by step.

### Part (1): Finding the Center of Symmetry of $ f(x) $

Given:
\[ f(x) = \sin^2 x + \sin x + \cos^2 x \]

First, simplify the function $ f(x) $:
\[ \cos^2 x + \sin^2 x = 1 \]

Thus,
\[ f(x) = 1 + \sin x \]

A function $ g(x) $ has a center of symmetry at $ x = a $ if $ g(a + x) = g(a - x) $.

For $ f(x) = 1 + \sin x $, we need to check if it has a center of symmetry.

Let $ a $ be the center of symmetry. Then:
\[ 1 + \sin(a + x) = 1 + \sin(a - x) \]

Since $ \sin $ is an odd function:
\[ \sin(a + x) = \sin(a - x) \]

For this equality to hold true, $ a $ must be a point where $ \sin $ is symmetrical, i.e., $ \sin(a) = 0 $.

Thus, the center of symmetry $ a $ is:
\[ a = k\pi, \, \text{where } k \in \mathbb{Z} \]

Therefore, the function $ f(x) = 1 + \sin x $ has centers of symmetry at:
\[ x = k\pi, \, \text{where } k \in \mathbb{Z} \]

### Part (2): Finding the Range of $ n $

Given:
\[ f(x) = 1 + \sin x \]

We need to find the number of zeros of $ f(x) $ in the interval $\left[ 0, \frac{n\pi}{2} \right]$.

The zeros of $ f(x) = 1 + \sin x $ occur when:
\[ 1 + \sin x = 0 \]
\[ \sin x = -1 \]

The sine function $\sin x = -1$ at:
\[ x = \frac{3\pi}{2} + 2k\pi, \, \text{where } k \in \mathbb{Z} \]

We need 2024 zeros in the interval:
\[ \left[ 0, \frac{n\pi}{2} \right] \]

Let’s count the number of such zeros:
\[ \frac{3\pi}{2} + 2k\pi \leq \frac{n\pi}{2} \]
\[ 3 + 4k \leq n \]
\[ 4k \leq n - 3 \]
\[ k \leq \frac{n - 3}{4} \]

Given there are 2024 zeros:
\[ k + 1 = 2024 \]
\[ k = 2023 \]

Thus:
\[ \frac{n - 3}{4} \geq 2023 \]
\[ n - 3 \geq 8092 \]
\[ n \geq 8095 \]

Therefore, the value of $ n $ must be:
\[ n \geq 8095 \]

### Summary
1. The centers of symmetry of the function $ f(x) = 1 + \sin x $ are $ x = k\pi $, where $ k \in \mathbb{Z} $.
2. The value of $ n $ such that the function $ f(x) $ has exactly 2024 zeros in the interval $\left[ 0, \frac{n\pi}{2} \right]$ is:
\[ n \geq 8095 \]

Do you want further details or have any questions?

Here are some related questions you might want to consider:
1. How do you determine the zeros of a trigonometric function?
2. What are the general properties of sine and cosine functions?
3. How do you solve trigonometric equations within a given interval?
4. What is the significance of symmetry in trigonometric functions?
5. How can you determine the period of a sine or cosine function?

**Tip:** When working with trigonometric functions, always remember their periodic nature and key values within one period. This can greatly simplify solving equations involving these functions.

Explore how to determine the center of symmetry and the range of n for which the function f(x) has exactly 2024 zeros. This comprehensive solution covers key trigonometric concepts suitable for Grades 10-12.

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