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Niclas Alexandersson edited a) Number of blits.tex
about 10 years ago
Commit id: c1d1bab720fa1928e95dadfaa6d9ca408e81f5ca
deletions | additions
diff --git a/a) Number of blits.tex b/a) Number of blits.tex
index d914a3f..f600f99 100644
--- a/a) Number of blits.tex
+++ b/a) Number of blits.tex
...
\begin{subequations}
\begin{align}
T(2^1) &= 5\\
T(2^2) &= 5 + 5
* \cdot 2^2\\
T(2^3) &= 5 + 5
* \cdot 2^2 + 5
* \cdot 2^4\\
T(2^4) &= 5 + 5
* \cdot 2^2 + 5
* \cdot 2^4 + 5
* \cdot 2^6\\
T(2^5) &= 5 + 5
* \cdot 2^2 + 5
* \cdot 2^4 + 5
* \cdot 2^6 + 5
* \cdot 2^8
\end{align}
\end{subequations}
...
T(2^n) &= \sum_{i = 0}^{n-1} \left[5 \cdot 4^i\right]
= 5\sum_{i = 0}^{n-1} \left[4^i\right]\\
&= 5\frac{1 - 4^n}{1 - 4}
= \frac{5 - 5
* \cdot 4^n}{-3}\\
&= \frac{5
* \cdot 4^n - 5}{3}
\end{split}
\end{equation}
Then:
\begin{equation}
\begin{split}
T(2^{n+1}) &= 5 + 4\frac{5
* \cdot 4^n - 5}{3}\\
&= 5 + \frac{5
* \cdot 4^{n+1} -
5*4}{3}\\ 5 \cdot 4}{3}\\
&= \frac{5
* \cdot 4^{n+1} + 5 * 3 - 5
* \cdot 4}{3}\\
&= \frac{5
* \cdot 4^{n+1} - 5}{3}
\end{split}
\end{equation}
Which gives us that: