Question

Please help me with this problem.

Answer 1

Easy way out:

1-3+9-27+81-243+729-2187 = -1640

Proper solution:

Sum of n terms of geometric:

S(n)=a((1-r^n)/(1-n)

a=1; r=-3; 2187=3^7 => n=8

S(8)=1(1-(-3)^8)/(1-(-3))=1(1-6561)/4=-1640

Answer 2

So, here's a cool "trick" we can use with geometric series:

`(-3)^0+(-3)^1+(-3)^2...+(-3)^7=S\\(-3)^1+(-3)^2+(-3)^3..(-3)^8=-3S\\-3S-S=(-3)^8-(-3)^0+((-3)^1-(-3)^1)+((-3)^2-(-3)^2)...\\-4S=(-3)^8-1\\S=((-3)^8-1)/(-4)=-1640`

We basically made the powers of -3 telescope here by first manipulating the first equation then subtracting it from the second.