Respuesta :
Answer:
Step-by-step explanation:
First solve for the common ratio:
[tex]\frac{30}{180} = \frac{1}{6} 6 \\ \\\frac{5}{30} = \frac{1}{6}[/tex]
Use the geometric summation equation to solve for sum:
[tex]a(\frac{(1-r^{n} )}{(1-r)})\\180(\frac{(1-\frac{1}{6} ^{6}) }{(1-\frac{1}{6} )}) = \frac{46655}{216} = 216[/tex]
The sum of the first 6 terms of the following series is 540.
What is the geometric series?
A geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms.
The sum of the terms of the geometric sequnece is given by;
[tex]\rm S_n=\dfrac{a(1-r^n)}{1-r}[/tex]
The common ratio of the sequence is;
[tex]\rm \dfrac{a_2}{a_1}=\dfrac{30}{180}=\dfrac{1}{6}\\\\ \dfrac{a_2}{a_1}=\dfrac{5}{30}=\dfrac{1}{6}\\\\[/tex]
The sum of the first 6 terms of the following series is;
[tex]\rm S_n=\dfrac{a(1-r^n)}{1-r}\\\\\rm S_6=\dfrac{180(1-1/6^6)}{1-1/6}\\\\ S_6=\dfrac{180\times 0.99}{5/6}\\\\S_6=\dfrac{180\times 0.99}{0.33}\\\\S_6=\dfrac{178.2}{0.33}\\\\S_6=540\\[/tex]
Hence, the sum of the first 6 terms of the following series is 540.
Learn more about geometric sequence here;
https://brainly.com/question/8934175
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