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The given recurrence relation is for the sequence \( u_n \), defined as:

\[
u_n = 2n^2 + 3n + 3
\]

We are asked to deduce the expression for \( u_{n+1} \) in terms of \( n \).

To find \( u_{n+1} \), we substitute \( n+1 \) into the formula for \( u_n \):

\[
u_{n+1} = 2(n+1)^2 + 3(n+1) + 3
\]

Let's simplify the expression:

\[
u_{n+1} = 2(n^2 + 2n + 1) + 3(n+1) + 3
\]

Expanding:

\[
u_{n+1} = 2n^2 + 4n + 2 + 3n + 3 + 3
\]

Now, combine like terms:

\[
u_{n+1} = 2n^2 + (4n + 3n) + (2 + 3 + 3)
\]

\[
u_{n+1} = 2n^2 + 7n + 8
\]

Thus, the expression for \( u_{n+1} \) is:

\[
u_{n+1} = 2n^2 + 7n + 8
\]

Would you like more details or clarification on any part of this process? 

Here are five related questions to explore further:
1. How do you find the general form for any recurrence relation?
2. What are some common methods to simplify quadratic expressions?
3. How does one prove that a sequence is arithmetic or geometric?
4. What is the difference between explicit and recursive formulas in sequences?
5. How do recurrence relations relate to polynomial growth in sequences?

**Tip**: When simplifying expressions with powers, always expand the terms carefully before combining like terms to avoid mistakes.

On considère la suite (u_n) définie explicitement par : u_n = 2n^2 + 3n + 3. En déduire l'expression de u_{n+1} en fonction de n.

Find the expression for u_{n+1} in terms of n for the sequence defined by u_n = 2n^2 + 3n + 3. This step-by-step solution simplifies the quadratic recurrence relation and explores key mathematical concepts. Suitable for Grades 10-12.

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