Question 878901
In a GP (a geometric progression or geometric sequence) each term is equal to the previous multiplied times a constant often called the common factor and represented by {{{r}}} .
 
MENTAL MATH (not showing the works of your quick mind):
The GP started with 5, and after multiplying times {{{r}}} four times we get
{{{5r^4=405=810/2=10*81/2*5=5*81=5*3^4}}} so {{{r=3}}} ,
which makes
{{{x=5*3=highlight(15)}}}
{{{x=15*3=highlight(45)}}}
{{{y=45*3=highlight(135)}}}
{{{z=135*3=highlight(405)}}}
 
SHOW YOUR WORK:
Teachers like to see formulas, and formulas are an efficient way to describe ideas and concepts. What you can think through in 5 seconds can be written in formulas in 5 minutes, while it would take much longer to write an explanation of the calculations in paragraph form. However, using formulas to communicate requires sharing a common symbolic language, and we may be used to different languages.
{{{b[1]}}}= first term of the GP.
{{{r}}}= common ratio of the GP.
{{{b[n]=b[1]r^(n-1)}}}={{{n^th}}}  term of the GP.
 
{{{b[1]=5}}}
{{{b[5]=405}}}
{{{5r^(5-1)=405}}}
{{{5r^4=405}}}
{{{r^4=405/5}}}
{{{r^4=81}}}
{{{r=root(4,81)}}}
{{{r=3}}}
{{{x=b[2]=5*3^(2-1)=5*3^1=5*3=highlight(15)}}}
{{{y=b[3]=5*3^(3-1)=5*3^2=5*9=highlight(45)}}}
{{{z=b[4]=5*3^(4-1)=5*3^3=5*27=highlight(135)}}}