The correct option for the question is C. [tex]S_{n}[/tex] = 15 [ 1 - [tex](3/4)^{n}[/tex] ]
What is a Geometric progression?
If in a sequence of terms, each succeeding term is generated by multiplying each preceding term with a constant value, then the series is referred to as a geometric progression. (GP), while the constant value is called the common ratio.
Conclusion: The nth term from the end of the GP with the last term l and common ratio r = l/ [r(n – 1)]. The sum of infinite, i.e. the sum of a GP with infinite terms is S∞= a/(1 – r) such that 0 < r < 1.
[tex]a_{n}[/tex] = 5[tex](3/4)^{n}[/tex]
[tex]S_{n}[/tex]= Σ[tex]a_{n}[/tex]
[tex]S_{n}[/tex] = ∑5[tex](3/4)^{n}[/tex]
= 5∑[tex](3/4)^{n}[/tex]
= 5 [ 3/4 + [tex](3/4)^{2}[/tex] + ......+ [tex](3/4)^{n}[/tex]]
=5 [ 3/4 * { 1- [tex](3/4)^{n}[/tex]} / 1 - 3/4 ]
[tex]S_{n}[/tex] = 15 [ 1 - [tex](3/4)^{n}[/tex] ]
learn more about geometric sequence here https://brainly.com/question/1509142
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