The geometric series' first seven terms are added together to form a sum equal to [tex]\boldsymbol{16383}[/tex].
A geometric series is the summation of an unlimited number of terms with a fixed ratio between them.
Let [tex]\boldsymbol{a,r}[/tex] denote the first term and the common ratio between the consecutive terms respectively.
Consider the geometric series: [tex]3+12+48+192+...[/tex]
[tex]\boldsymbol{a=3}[/tex]
[tex]r=\frac{12}{3}[/tex]
[tex]= \boldsymbol{4}[/tex]
[tex]\boldsymbol{S_n=\frac{a(1-r^n)}{1-r} }[/tex]
[tex]S_n =\frac{3(1-4^7)}{1-4}[/tex]
[tex]=\frac{3(16383)}{3}[/tex]
[tex]=16383[/tex]
So, the geometric series' first seven terms are added together to form a sum equal to [tex]\boldsymbol{16383}[/tex].
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