To solve this problem, we will follow the iteration process step by step as outlined by the given formula [tex]\( x_{n+1} = \frac{7 - x_n^3}{8} \)[/tex].
Given:
[tex]\( x_1 = 2.9 \)[/tex]
We need to find [tex]\( x_3 \)[/tex] and [tex]\( x_4 \)[/tex] with each answer rounded to 1 decimal place.
### Step-by-Step Solution:
1. Calculate [tex]\( x_2 \)[/tex] using [tex]\( x_1 \)[/tex]:
[tex]\[
x_2 = \frac{7 - x_1^3}{8}
\][/tex]
Substituting [tex]\( x_1 = 2.9 \)[/tex]:
[tex]\[
x_2 = \frac{7 - (2.9)^3}{8}
\][/tex]
2. Calculate [tex]\( x_3 \)[/tex] using [tex]\( x_2 \)[/tex]:
[tex]\[
x_3 = \frac{7 - x_2^3}{8}
\][/tex]
Substituting the calculated value of [tex]\( x_2 \)[/tex]:
[tex]\[
x_3 = \frac{7 - (-2.173625)^3}{8}
\][/tex]
3. Calculate [tex]\( x_4 \)[/tex] using [tex]\( x_3 \)[/tex]:
[tex]\[
x_4 = \frac{7 - x_3^3}{8}
\][/tex]
Substituting the calculated value of [tex]\( x_3 \)[/tex]:
[tex]\[
x_4 = \frac{7 - (2.1587009850754395)^3}{8}
\][/tex]
4. Rounding to 1 decimal place:
- The value of [tex]\( x_3 \)[/tex] calculated is approximately 2.1587009850754395, which rounds to 2.2.
- The value of [tex]\( x_4 \)[/tex] calculated is approximately -0.3824406100402049, which rounds to -0.4.
### Conclusion:
- [tex]\( x_3 \)[/tex] is approximately 2.2 when rounded to 1 decimal place.
- [tex]\( x_4 \)[/tex] is approximately -0.4 when rounded to 1 decimal place.
