Answer:
[tex]S_{75}=-506.888989761$ X 10^{32}[/tex]
Step-by-step explanation:
Given the sequence
[tex]a_1=-\dfrac{1}{3}\\ a_i=a_{i-1}\cdot (-3)[/tex]
[tex]a_2=a_{2-1}\cdot (-3)=a_1 \cdot (-3) = -\dfrac{1}{3}\cdot (-3)=1\\a_3=a_{3-1}\cdot (-3)=a_2 \cdot (-3) = 1 \cdot (-3)=-3\\a_4=a_{4-1}\cdot (-3)=a_3 \cdot (-3) = -3 \cdot (-3)=9\\[/tex]
Therefore the sequence is:
[tex]-\dfrac{1}{3}, 1, -3, 9, ...[/tex]
This is a geometric sequence where the:
First Term, [tex]a_1=-\dfrac{1}{3}[/tex]
Common ratio, r =-3
We want to determine the sum of the first 75 terms.
For a geometric sequence, the sum:
[tex]S_n=\dfrac{a_1(1-r^n)}{1-r} \\[/tex]
Therefore:
[tex]S_{75}=\dfrac{-\dfrac{1}{3}(1-(-3)^{75})}{1-(-3)} \\\\=\dfrac{-(1-(-3)^{75})}{4*3}\\\\S_{75}=\dfrac{(-3)^{75}-1}{12}\\\\S_{75}=-506.888989761$ X 10^{32}[/tex]
