Question 1104788
use the principals of mathematical induction to prove the following statement 1+5+5^2+...+5^n-1 = 1/4(5n-1) 

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The right hand side  is not correct.   It should be {{{ (1/4)(5^n - 1) }}} 
Let {{{ S[n] }}} = partial sum n  (where {{{5^(n-1) }}} is the highest power in the sum)

Proof by induction:
{{{ S[1] = 1 = (1/4)(5^1 - 1) }}}
{{{ S[2]  = 6 = (1/4)(5^2 - 1) }}}
Assume {{{ S[k] = 1+5+5^2  }}} + … + {{{ 5^(k-1) }}} = {{{ (1/4)(5^k - 1) }}}  is true for n=k.

Now we need to show that that assumption leads to  {{{ S[k+1] = (1/4)(5^(k+1) - 1 ) }}} :

{{{ S[k+1] = 1+5+5^2 }}} + … + {{{ 5^(k-1) + 5^k }}}   
On the right hand side, all but the last term are really just {{{ S[k] }}}
{{{ S[k+1] = S[k] + 5^k }}}
{{{ S[k+1] = (1/4)(5^k - 1) + 5^k  }}}
{{{ S[k+1] = (1/4)(5^k - 1 + 4*5^k) }}}
{{{ S[k+1] = (1/4)(5*5^k - 1) }}}
{{{ S[k+1] = (1/4)(5^(k+1) - 1) }}}  
 DONE.