Question

Question #ca917

Answer 1

detail in explanation

Explanation:

1. `n=1`
   `1=1/4(5^1-1)` correspond
2. let `n=k`
   `1+5+5^2+5^3+...+5^(k-1)=1/4(5^k-1)` is true
3. 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 , `1+5+5^2+...+5^(n-1)=1/4(5^n-1)`
is true for all positive integer n