EXPLANATION
[tex]\mathrm{A\: geometric\: sequence\: has\: a\: constant\: ratio\: }r\mathrm{\: and\: is\: defined\: by}\: a_n=a_0\cdot r^{n-1}[/tex]Compute the ratios of all the adjacent terms: r=(a_n+1)/(a_n)
[tex]\frac{16384}{32768}=\frac{1}{2},\: \quad \frac{8192}{16384}=\frac{1}{2}[/tex][tex]\mathrm{The\: ratio\: of\: all\: the\: adjacent\: terms\: is\: the\: same\: and\: equal\: to}[/tex][tex]r=\frac{1}{2}[/tex][tex]\mathrm{The\: first\: element\: of\: the\: sequence\: is}[/tex][tex]a_1=32768[/tex][tex]a_n=a_1\cdot r^{n-1}[/tex][tex]\mathrm{Therefore,\: the\: }n\mathrm{th\: term\: is\: computed\: by}\: [/tex][tex]r=\frac{1}{2},\: a_n=32768\mleft(\frac{1}{2}\mright)^{n-1}[/tex]Hence, the 13th term would be:
[tex]a_{13}=32768(\frac{1}{2})^{13-1}=32768(\frac{1}{2})^{12}=32768\cdot\frac{1}{4096}=8[/tex]The answer is n_13=8
