Let's simplify the expression step by step:
[tex]\[
\frac{i + 2i^2 + 3i^3 + 4i^4 + 5i^5 + 6i^6 + 7i^8}{8i^9 + 9i^{10} + 10i^{11} + 11i^{12} + 12i^{13}}
\][/tex]
First, recall the fundamental powers of the imaginary unit [tex]\(i\)[/tex]:
[tex]\[
\begin{align*}
i^1 &= i \\
i^2 &= -1 \\
i^3 &= -i \\
i^4 &= 1
\end{align*}
\][/tex]
These values repeat every four powers. Using this periodicity, we can express higher powers in terms of [tex]\(i, -1, -i,\)[/tex] and 1.
Let's simplify the numerator:
[tex]\[
\begin{align*}
i &= i \\
2i^2 &= 2(-1) = -2 \\
3i^3 &= 3(-i) = -3i \\
4i^4 &= 4(1) = 4 \\
5i^5 &= 5(i) = 5i \\
6i^6 &= 6(-1) = -6 \\
7i^8 &= 7(1) = 7
\end{align*}
\][/tex]
So, the numerator becomes:
[tex]\[
i - 2 - 3i + 4 + 5i - 6 + 7
\][/tex]
Combine like terms:
[tex]\[
(i - 3i + 5i) + (-2 + 4 - 6 + 7) = (3i) + (3)
\][/tex]
Thus, the simplified numerator is:
[tex]\[
3 + 3i
\][/tex]
Next, let's simplify the denominator:
[tex]\[
\begin{align*}
8i^9 &= 8i \\
9i^{10} &= 9i^2 = 9(-1) = -9 \\
10i^{11} &= 10i^3 = 10(-i) = -10i \\
11i^{12} &= 11(1) = 11 \\
12i^{13} &= 12i
\end{align*}
\][/tex]
So, the denominator becomes:
[tex]\[
8i - 9 - 10i + 11 + 12i
\][/tex]
Combine like terms:
[tex]\[
(8i - 10i + 12i) + (-9 + 11) = 10i + 2
\][/tex]
Thus, the simplified denominator is:
[tex]\[
2 + 10i
\][/tex]
Now we have:
[tex]\[
\frac{3 + 3i}{2 + 10i}
\][/tex]
To simplify the fraction further, we multiply the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate of [tex]\(2 + 10i\)[/tex] is [tex]\(2 - 10i\)[/tex]:
[tex]\[
\frac{(3 + 3i)(2 - 10i)}{(2 + 10i)(2 - 10i)}
\][/tex]
Calculate the numerator:
[tex]\[
(3 + 3i)(2 - 10i) = 6 - 30i + 6i - 30i^2 = 6 - 24i + 30 = 36 - 24i
\][/tex]
And the denominator:
[tex]\[
(2 + 10i)(2 - 10i) = 4 - 100i^2 = 4 + 100 = 104
\][/tex]
So the fraction simplifies to:
[tex]\[
\frac{36 - 24i}{104}
\][/tex]
Which can further be simplified by dividing both the real and the imaginary components by 104:
[tex]\[
\frac{36}{104} - \frac{24i}{104} = \frac{9}{26} - \frac{3i}{13}
\][/tex]
Therefore, the fully simplified expression is:
[tex]\[
\frac{9}{26} - \frac{3i}{13}
\][/tex]
