Answer:
-1640
Explanation:
You want the sum of the geometric series ...
1 -3 + ... -2187
Series
We can write out this series with first term 1 and common ratio -3:
1 -3 +9 -27 +81 -243 +729 -2187
And, we can add it up:
1 -3 +9 -27 +81 -243 +729 -2187 = -1640
The sum of the series is -1640.
Formulas
The formula for the general term of the series is ...
an = a1(r^(n-1))
We can use this to find the number (n) of the last term:
-2187 = 1((-3)^(n-1))
If we consider the magnitudes of the terms, the value of n is the same:
2187 = 1(3^(n-1))
3^7 = 3^(n -1)
7 = n -1
8 = n . . . . . . . the same number we get by counting terms in the series
And the formula for the sum is ...
Sn = a1(r^n -1)/(r -1)
S8 = 1((-3)^8 -1)/(-3-1) = (6561 -1)/-4 = -1640
The sum of the series is -1640.
__
Additional comment
As we did above, we would ordinarily find the value of n using logarithms. The logs of negative numbers are complex.
In order to get a real ratio of ln(-2187) to ln(-3), we need to adjust the imaginary part of ln(-2187) by multiples of 2π so the result has the same angle as ln(-3). What we find is ...
(ln(-2187) +6πi)/(ln(-3)) = 7 . . . . a positive integer
Effectively, we can find n by considering only the real parts of the logs, which is basically the same as using term magnitudes.