Question

What is the next term in the geometric sequence 16, –4, 1, –(1/4), … ?

Answer 1

`a_1=16;\ a_2=-4;\ a_3=1;\ a_4=-(1)/(4)\\\\q=a_2:a_1\\\\q=-4:16=-(1)/(4)\\\\a_n=a_1\cdot q^(n-1)\\\\a_n=16\cdot\left(-(1)/(4)\right)^(n-1)=16\cdot\left(-(1)/(4)\right)^n\cdot\left(-(1)/(4)\right)^(-1)\\\\=16\cdot\left(-(1)/(4)\right)^n\cdot(-4)=-64\cdot\left(-(1)/(4)\right)^n=(-4)^3\cdot\left(-(1)/(4)\right)^n=\left(-(1)/(4)\right)^(-3)\cdot\left(-(1)/(4)\right)^n\\\\=\left(-(1)/(4)\right)^(n-3)`


`a_5=\left(-(1)/(4)\right)^(5-3)=\left(-(1)/(4)\right)^2=(1)/(16)\\\\a_6=\left(-(1)/(4)\right)^(6-3)=\left(-(1)/(4)\right)^3=-(1)/(64)\\\vdots`