Objective:
To prove #Sigma_(k=1)^(n) ar^(k-1) = a(1-r^n)/(1-r)#
#"----------------------------------------------------------------------------"#
Show that the equation is true for #color(black)(n=1)#
If #color(black)(n=1)#
#Sigma_(k=1)^1 ar^(k-1) = ar^0 = a = a(1-r^1)/(1-r)#
#"----------------------------------------------------------------------------"#
Show that if it is true for #color(black)(n-1)# then it is true for #color(black)(n)#
Assuming #color(red)(Sigma_(k=1)^(n-1) ar^(k-1)=a(1-r^(n-1))/(1-r))#
#Sigma_(k=1)^n ar^(k-1) = color(red)(Sigma_(k=1)^(n-1)ar^(k-1))+color(blue)(ar^(n-1))#
#color(white)("XXXXXX")= a(1-r^(n-1))/(1-r) +a(r^(n-1)(1-r))/(1-r)#
#color(white)("XXXXXX")=a(1-r^(n-1)+r^(n-1)-r^n)/(1-r)#
#color(white)("XXXXXX")=a(1-r^n)/(1-r)#
#"======================================================="#
