Question #ca917
1 Answer
May 4, 2017
detail in explanation
Explanation:
#n=1 #
#1=1/4(5^1-1)# correspond- let
#n=k#
#1+5+5^2+5^3+...+5^(k-1)=1/4(5^k-1)# is true - if
#n=k+1#
#1+5+5^2+5^3+...+5^(k-1)+5^(k+1-1)=1/4(5^k-1)+5^k#
then observe the right pattern temporarily
#=>1/4*5^k+5^k-1/4#
#=>(5/4)*5^k-1/4#
#=>(1/4)*5*5^k-1/4#
#=>1/4*(5^(k+1)-1)#
so
#1+5+5^2+5^3+...+5^(k-1)+5^(k+1-1)=1/4*(5^(k+1)-1)#
is also true
conclusion:
by mathematical induction ,
is true for all positive integer n