Question

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

Answer 1

$a_1=16;\ a_2=-4;\ a_3=1;\ a_4=-\frac{1}{4}\\\\q=a_2:a_1\\\\q=-4:16=-\frac{1}{4}\\\\a_n=a_1\cdot q^{n-1}\\\\a_n=16\cdot\left(-\frac{1}{4}\right)^{n-1}=16\cdot\left(-\frac{1}{4}\right)^n\cdot\left(-\frac{1}{4}\right)^{-1}\\\\=16\cdot\left(-\frac{1}{4}\right)^n\cdot(-4)=-64\cdot\left(-\frac{1}{4}\right)^n=(-4)^3\cdot\left(-\frac{1}{4}\right)^n=\left(-\frac{1}{4}\right)^{-3}\cdot\left(-\frac{1}{4}\right)^n\\\\=\left(-\frac{1}{4}\right)^{n-3}$

$a_5=\left(-\frac{1}{4}\right)^{5-3}=\left(-\frac{1}{4}\right)^2=\frac{1}{16}\\\\a_6=\left(-\frac{1}{4}\right)^{6-3}=\left(-\frac{1}{4}\right)^3=-\frac{1}{64}\\\vdots$