Respuesta :
Step-by-step explanation:
Given the geometric sequence
8 + 6 + 4.5...
A geometric sequence has a constant ratio and is defined by
[tex]a_n=a_1\cdot r^{n-1}[/tex]
[tex]\mathrm{Compute\:the\:ratios\:of\:all\:the\:adjacent\:terms}:\quad \:r=\frac{a_{n+1}}{a_n}[/tex]
[tex]\frac{6}{8}=\frac{3}{4},\:\quad \frac{4.5}{6}=\frac{3}{4}[/tex]
[tex]\mathrm{The\:ratio\:of\:all\:the\:adjacent\:terms\:is\:the\:same\:and\:equal\:to}[/tex]
[tex]r=\frac{3}{4}[/tex]
[tex]\mathrm{The\:first\:element\:of\:the\:sequence\:is}[/tex]
[tex]a_1=8[/tex]
[tex]\mathrm{Therefore,\:the\:}n\mathrm{th\:term\:is\:computed\:by}\:[/tex]
[tex]a_n=8\left(\frac{3}{4}\right)^{n-1}[/tex]
[tex]\mathrm{Geometric\:sequence\:sum\:formula:}[/tex]
[tex]a_1\frac{1-r^n}{1-r}[/tex]
[tex]\mathrm{Plug\:in\:the\:values:}[/tex]
[tex]n=25,\:\spacea_1=8,\:\spacer=\frac{3}{4}[/tex]
[tex]=8\cdot \frac{1-\left(\frac{3}{4}\right)^{25}}{1-\frac{3}{4}}[/tex]
[tex]\mathrm{Multiply\:fractions}:\quad \:a\cdot \frac{b}{c}=\frac{a\:\cdot \:b}{c}[/tex]
[tex]=\frac{\left(1-\left(\frac{3}{4}\right)^{25}\right)\cdot \:8}{1-\frac{3}{4}}[/tex]
[tex]=\frac{8\left(-\left(\frac{3}{4}\right)^{25}+1\right)}{\frac{1}{4}}[/tex]
[tex]\mathrm{Apply\:exponent\:rule}:\quad \left(\frac{a}{b}\right)^c=\frac{a^c}{b^c}[/tex]
[tex]=\frac{8\left(-\frac{3^{25}}{4^{25}}+1\right)}{\frac{1}{4}}[/tex]
[tex]\mathrm{Apply\:the\:fraction\:rule}:\quad \frac{a}{\frac{b}{c}}=\frac{a\cdot \:c}{b}[/tex]
[tex]=\frac{\left(1-\frac{3^{25}}{4^{25}}\right)\cdot \:8\cdot \:4}{1}[/tex]
[tex]\mathrm{Multiply\:the\:numbers:}\:8\cdot \:4=32[/tex]
[tex]=\frac{32\left(-\frac{3^{25}}{4^{25}}+1\right)}{1}[/tex]
[tex]=\frac{32\cdot \frac{4^{25}-3^{25}}{4^{25}}}{1}[/tex] ∵ [tex]\mathrm{Join}\:1-\frac{3^{25}}{4^{25}}:\quad \frac{4^{25}-3^{25}}{4^{25}}[/tex]
[tex]=32\cdot \frac{4^{25}-3^{25}}{4^{25}}[/tex]
[tex]=\frac{\left(4^{25}-3^{25}\right)\cdot \:32}{4^{25}}[/tex]
[tex]=\frac{2^5\left(4^{25}-3^{25}\right)}{2^{50}}[/tex] ∵ [tex]\mathrm{Factor}\:32:\ 2^5[/tex], [tex]\mathrm{Factor}\:4^{25}:\ 2^{50}[/tex]
so
[tex]=\frac{4^{25}-3^{25}}{2^{45}}[/tex] ∵ [tex]\mathrm{Cancel\:}\frac{\left(4^{25}-3^{25}\right)\cdot \:2^5}{2^{50}}:\quad \frac{4^{25}-3^{25}}{2^{45}}[/tex]
[tex]\mathrm{Apply\:the\:fraction\:rule}:\quad \frac{a\pm \:b}{c}=\frac{a}{c}\pm \frac{b}{c}[/tex]
[tex]=\frac{4^{25}}{2^{45}}-\frac{3^{25}}{2^{45}}[/tex]
[tex]=32-\frac{3^{25}}{2^{45}}[/tex] ∵ [tex]\frac{4^{25}}{2^{45}}=32[/tex]
[tex]=32-0.024[/tex] ∵ [tex]\frac{3^{25}}{2^{45}}=0.024[/tex]
[tex]=31.98[/tex]
Therefore, the sum of the first 25 terms in this geometric series: 31.98
