MATEMATIKA OLIMPIADE: 3 deret geometri tak hingga yang dijumlahkan

Hitung :
$\small \fn_jvn \sum_{k=1}^{\infty }\frac{4^{k}+3^{k-1}-2^{k-2^{}}}{5^{k+1}}$

Jawab :
$\small \fn_jvn \sum_{k=1}^{\infty }\frac{4^{k}+3^{k-1}-2^{k-2^{}}}{5^{k+1}}$
$\small \fn_jvn =\sum_{k=1}^{\infty }(\frac{4^{k}}{5^{k+1}}+\frac{3^{k-1}}{5^{k+1}}-\frac{2^{k-2}}{5^{k+1}})$
$\small \fn_jvn =\sum_{k=1}^{\infty }\left ( \frac{1}{5}\left ( \frac{4}{5} \right )^{k}+\frac{1}{15}\left ( \frac{3}{5} \right )^{k}-\frac{1}{20}\left (\frac{2}{5} \right )^{k} \right )$
$\small \fn_jvn =\frac{1}{5}\left ( \frac{\frac{4}{5}}{1-\frac{4}{5}}+\frac{1}{15}\left ( \frac{\frac{3}{5}}{1-\frac{3}{5}} \right )-\frac{1}{20}\left ( \frac{\frac{2}{5}}{1-\frac{2}{5}} \right ) \right )=\frac{13}{15}$

MATEMATIKA OLIMPIADE

Wednesday, September 7, 2011

3 deret geometri tak hingga yang dijumlahkan

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