The sum of the first 75 terms of the sequence is -5.068*10^34
Given the following geometric sequence
[tex]a_1 = \frac{-1}{3} \\a_i = a_{i-1}*(-3)[/tex]
The sum of the nth term of the GP is expressed as;
[tex]S_n = \frac{a(1-r^n)}{1-r}[/tex]
a = -1/3
n = 75
r = a₂/a₁
a₂ = -3a₁
a₂ = -3(-1/3)
a2 = 1
r = 1/(-1/3)
r = -3
Substitute the given values in the formula;
[tex]S_n = \frac{-1/3(1-3^{75})}{1-(-3)}\\S_{75} = \frac{-1/3(3^{75}-1)}{1+3} \\S_{75} = \frac{-1/3(6.082 \times 10^{35}-1)}{4}\\S_{75} = \frac{-1/3(6.082 \times 10^{35})}{4}\\S_{75} = -0.5068 \times 10^{35}\\S_{75} = -5.068 \times 10^{34}\\[/tex]
Hence the sum of the first 75 terms is -5.068*10^34
Learn more on geometric series here: https://brainly.com/question/20548958
