Question #ca917

1 Answer
May 4, 2017

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