Question

Find an equation for the nth term of a geometric sequence where the second and fifth terms are -8 and 512, respectively.

Answer 1

Easy
recall that
an=a1(r)^(n-1)
so

given 2nd and 5th term

we get
a2 and a5
so
a2=a1(r)^(2-1)=a1(r)^1=a1r
a5=a1(r)^(5-1)=a1(r)^4

also remember that $\frac{x^m}{x^n}=x^{m-n}$
so
$\frac{a_5}{a_2}= \frac{a_1r^4}{a_1r^1} =r^{4-1}=r^3= \frac{512}{-8}=-64$
so r^3=-64
cube root
r=-4
so

a2=a1r=-8
a2=a1(-4)=-8
divide both sides by -4
a1=2

so

equation is
$a_n=2(-4)^{n-1}$

C isi the answer