Answer:
n = 8
Step-by-step explanation:
The given sequence, 2, 6, 18, 54. . ., is a geometric sequence.
It has a common ratio of 3 => [tex] \frac{6}{2} = \frac{18}{6} = \frac{54}{18} = 3 [/tex]
Thus, the sum of the first n terms of a geometric sequence is given as [tex]S_n = \frac{a_1(1 - r^n)}{1 - r}[/tex]
Where,
[tex] a_1 [/tex] = first term of the series = 2
r = common ratio = 3
[tex] S_n [/tex] = sum of the first n terms = 6,560
Plug in the above values into the formula
[tex]6,560 = \frac{2(1 - 3^n)}{1 - 3}[/tex]
[tex] 6,560 = \frac{2(1 - 3^n)}{-2} [/tex]
[tex] 6,560 = \frac{1 - 3^n}{-1} [/tex]
Multiply both sides by -1
[tex] -6,560 = 1 - 3^n [/tex]
Subtract 1 from both sides
[tex] -6,560 - 1 = - 3^n [/tex]
[tex] -6,561 = - 3^n [/tex]
[tex] 6,561 = 3^n [/tex]
Evaluate
[tex] 3^8 = 3^n [/tex]
3 cancels 3
[tex] 8 = n [/tex]
The value of n = 8
