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Evaluate:

[tex]\bf{\sum^{20}_1\:4(\cfrac{8}{9})^{n-1}[/tex]



(find the sum of the first 20 terms of the geometric series)



Please help A.S.A.P. & show work as well.


Thank you guys!


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Sagot :

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Answer:   [tex]\text \large S_{20} \ = \sf \dfrac{4\left(9^{20}-8^{20}\right)}{3^{38}} \ \ \ or \ \ \ 32.586[/tex]

Given expression:

[tex]\boxed{\sf \sum _{n=1}^{20}\:4\left(\frac{8}{9}\right)^{n-1}}[/tex]

       Identify the following:

  • First Term (a) = 4(8/9)¹⁻¹ = 4

  • Common ratio (r) = 8/9

  • Total Terms (n) = 20

Formula Required:

[tex]\rightarrow \quad \sf S_n = \dfrac{a(r^n - 1)}{r-1}[/tex]

Insert values identified:

[tex]\rightarrow \sf S_{20} = \dfrac{4(\dfrac{8}{9}^{20} - 1)}{\dfrac{8}{9} -1} \quad\overset{simplify}{\longrightarrow} \quad \dfrac{4\left(9^{20}-8^{20}\right)}{3^{38}} \quad \xrightarrow{\text{In \ Decimals} }\quad 32.58609013[/tex]

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