We are given a recursive sequence defined by the term-to-term rule:

$u_{n+1} = u_n^2 - 3u_n + 1$

with the initial term $u_1 = 3$. Let's solve step-by-step:

a) Find $u_2$:

Using $u_1 = 3$:
$u_2 = u_1^2 - 3u_1 + 1$ Substitute $u_1 = 3$:
$u_2 = 3^2 - 3(3) + 1$ $u_2 = 9 - 9 + 1 = 1$

b) Find $u_3$:

Using $u_2 = 1$:
$u_3 = u_2^2 - 3u_2 + 1$ Substitute $u_2 = 1$:
$u_3 = 1^2 - 3(1) + 1$ $u_3 = 1 - 3 + 1 = -1$

c) Find $u_4$:

Using $u_3 = -1$:
$u_4 = u_3^2 - 3u_3 + 1$ Substitute $u_3 = -1$:
$u_4 = (-1)^2 - 3(-1) + 1$ $u_4 = 1 + 3 + 1 = 5$

Final Answers:

a) $u_2 = 1$
b) $u_3 = -1$
c) $u_4 = 5$

Would you like further explanations or have any other questions?

Here are 5 related questions to expand your understanding:

1. What happens to the sequence for larger values of $n$?
2. Can this recursive sequence converge? If so, what would the limit be?
3. How does changing the initial value $u_1$ affect the sequence?
4. Can you graph the first few terms of this sequence?
5. Is it possible to generalize $u_n$ explicitly without recursion?

Tip: Always verify recursive sequences by substituting back a few terms to check for calculation errors!