Question

What is the sum of the first 11 terms of the geometric sequence: -4, 12, -36, 108,...

Answer 1

`a_1=-4,\ a_2=12,\ a_3=-36,\ a_4=108,\ ...\\\\a_2:a_1=12:(-4)=-3\\a_3:a_2=-36:12=-3\\a_4:a_3=108:(-36)=-3\\a_(n+1):a_n=-3=const.\\\\\text{It's a geometric sequence}\\\\a_n=a_1r^(n-1)\to a_n=-4\cdot(-3)^(n-1)=-4\cdot(-3)^(-1)\cdot(-3)^n\\\\=-4\cdot\left(-(1)/(3)\right)\cdot(-3)^n=(4)/(3)(-3)^n\\\\\text{The formula of a Sum of the First n Terms of a Geometric Sequence:}`

`S_n=(a_1(1-r^n))/(1-r)\\\\\text{We have:}\\a_1=-4,\ r=-3,\ n=11\\\\\text{Substitute:}\\\\S_(11)=(-4[1-(-3)^(11)])/(1-(-3))=(-4(1+177147))/(4)=-177148\\\\Answer:\ -177,148`