Question 1141229
please help me with prove of mathematical induction of this question 1+r+r^2+r^3+....+r^n =1-r^n+1÷1-r.
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Please help me prove by mathematical induction:
  {{{1+r+r^2+r^3}}}+....+ {{{r^n  = (1-r^(n+1))/(1-r) }}}

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NOTE that  {{{ (1-r^(n+1)) / (1-r) = (r^(n+1)-1) / (r-1) }}}  ({{{r <> 1}}} of course)


n=1:  LHS:  {{{ 1 + r^1 = r+1  }}} 
      RHS:  {{{ (r^(1+1)-1)/(r-1) = (r^2-1)/(r-1) = r+1 }}}   (ok for n=1)

n=k:  Hypothesis is:  {{{ 1+r^1+r^2 }}} +...+ {{{ r^k }}} = {{{ (r^(k+1)-1)/(r-1) }}}


Let n=k+1:  Must show LHS for n=k+1 leads to {{{ (r^(k+2)-1)/(r-1) }}}:

     LHS:  {{{ 1+r^1+r^2 }}} +...+ {{{ r^k + r^(k+1) }}}
 
           Apply hypothesis to all but {{{r^(k+1) }}} term:
         = {{{ (r^(k+1)-1)/(r-1) + r^(k+1) }}}   

           Put everything over {{{r-1}}}:
         = {{{ (r^(k+1)-1)/(r-1) + (r^(k+1)(r-1))/(r-1) }}} 

           Simplifying (showing steps):
         = {{{ (r^(k+1)-1)/(r-1) + (r^(k+2)-r^(k+1))/(r-1) }}}    

         = {{{ ((r^(k+1)-1) + (r^(k+2)-r^(k+1)))/(r-1) }}}    

         = {{{ ((r^(k+2)-1))/(r-1) }}}    

     DONE.   We've shown assuming truth for n=k  implies   truth for n=k+1.