[tex]\large \bold {SOLUTION}[/tex]
[tex]\small\textsf{Use the sum of geometric sequence formula} \\ \\ \sf\blue{S_{n} = \dfrac{a_{1}(1 - {r}^{n} )}{1 - r} }[/tex]
[tex]\small\sf\red{S_{n} \: is \: the \: sum \: of \: the \: geometric \: sequence}[/tex]
[tex]\small\sf\red{a_{1} \: is \: the \: first \: term \: of \: the \: geometric \: sequence}[/tex]
[tex]\small\textsf\red{n is the number of terms in the sequence}[/tex]
[tex]\small\textsf\red{r is the geometric ratio of the sequence}[/tex]
[tex]\small\sf{Let \: S_{n} = 524288, \: a_{1} = 2 \: and \: r = 4}[/tex]
[tex]\small\sf{S_{n} = \dfrac{a_{1}(1 - {r}^{n} )}{1 - r} }[/tex]
[tex]\small\sf{524288 = \dfrac{2(1 - {4}^{n} )}{1 - 4} }[/tex]
[tex]\small\sf{524288 = \dfrac{2(1 - {4}^{n} )}{ - 3} }[/tex]
[tex]\small\sf{524288 = \dfrac{2 - (2)({4}^{n} )}{ - 3} }[/tex]
[tex]\small\sf{524288 = \dfrac{2 - (2 ) ^{1} ({2 }^{2n} )}{ - 3} }[/tex]
[tex]\small\sf{524288 = \dfrac{2 - ({2 })^{2n + 1} }{ - 3} }[/tex]
[tex]\small\sf{524288( - 3) = (\dfrac{2 - ({2 })^{2n + 1} }{\cancel{ - 3}} )(\cancel{ - 3})}[/tex]
[tex]\small\sf{ - 1572864 = 2 - ({2 })^{2n + 1}}[/tex]
[tex]\small\sf{ - 1572864 + 1572864 + ({2 })^{2n + 1}= 2 - ({2 })^{2n + 1} + 1572864 + ({2 })^{2n + 1}}[/tex]
[tex]\small\sf{ {2}^{2n + 1} = 1572864}[/tex]
[tex]\small\textsf{To extract n in the exponent, use logarithms}[/tex]
[tex]\small\sf{2n - 1 = log_{2}(1572864) }[/tex]
[tex]\small\sf{2n - 1 + 1= log_{2}(1572864) + 1 }[/tex]
[tex]\small\sf{2n = log_{2}(1572864) + 1 }[/tex]
[tex]\small\sf{ \dfrac{\cancel{2}n}{\cancel{2}} = \dfrac{ log_{{2}}(1572864) + 1 }{2} }[/tex]
[tex]\small\boxed{\green{\sf{n≈10 }}}[/tex]
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