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Given the sequence defined by [tex] u_{n+1} = u_n^3 + 2 [/tex] and the initial term [tex] u_0 = -3 [/tex]:

a) Find [tex] u_1 [/tex].

b) Find [tex] u_2 [/tex].

Sagot :

GinnyAnswer
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].
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