https://r2.erweima.ai/mathbot/image/1725794975099pu53cmm0.jpg

Let's break down the tasks based on the image. The image appears to be a mathematical problem involving sequences and Python programming. Here is a summary of what is asked:

### 1. Conjectures (about a recursive sequence)
The sequence \( (u_n) \) is defined by the recurrence relation:
\[ u_0 = 1 \]
\[ u_{n+1} = 2u_n - 3 \quad \text{for all integers } n \geq 0 \]

The tasks are:
(a) Using a calculator, a program, or a spreadsheet, calculate the first six terms of the sequence when \( u_0 = 1 \).
(b) Make conjectures about the variation of the sequence when \( u_0 = 1 \), then \( u_0 = 6 \), and finally \( u_

Conjectures: A sequence is defined by its first term u_0 = 1 and the recurrence relation u_{n+1} = 2u_n - 3 for all integers n >= 0. Tasks include calculating the first six terms of the sequence for u_0 = 1, observing variations, making conjectures for u_0 = 6, performing calculations for altered values of u_0, and interpreting the results of a Python function that iterates on a similar recursive process.

Explore the conjectures related to a sequence defined by the recurrence relation u_{n+1} = 2u_n - 3. Learn how to calculate the first six terms, observe variations for different initial values, and interpret Python code analyzing a similar recursive process. Ideal for high school students in Grades 10-12.

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