Let's solve this step-by-step.
We are given the initial value [tex]\( u_0 = -3 \)[/tex] and the recurrence relation [tex]\( u_{n+1} = u_n^3 + 2 \)[/tex].
### Part (a): Find [tex]\( u_1 \)[/tex]
To find [tex]\( u_1 \)[/tex], substitute [tex]\( u_0 \)[/tex] into the recurrence relation:
[tex]\[ u_1 = u_0^3 + 2 \][/tex]
Given [tex]\( u_0 = -3 \)[/tex]:
[tex]\[ u_1 = (-3)^3 + 2 \][/tex]
[tex]\[ u_1 = -27 + 2 \][/tex]
[tex]\[ u_1 = -25 \][/tex]
So, [tex]\( u_1 = -25 \)[/tex].
### Part (b): Find [tex]\( u_2 \)[/tex]
Next, to find [tex]\( u_2 \)[/tex], we use the value of [tex]\( u_1 \)[/tex] in the recurrence relation:
[tex]\[ u_2 = u_1^3 + 2 \][/tex]
Given [tex]\( u_1 = -25 \)[/tex]:
[tex]\[ u_2 = (-25)^3 + 2 \][/tex]
[tex]\[ u_2 = -15625 + 2 \][/tex]
[tex]\[ u_2 = -15623 \][/tex]
So, [tex]\( u_2 = -15623 \)[/tex].
