SOLUTION: 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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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.
Answer by math_helper(2461) About Me  (Show Source):
You can put this solution on YOUR website!
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%2Br%2Br%5E2%2Br%5E3+....+ r%5En++=+%281-r%5E%28n%2B1%29%29%2F%281-r%29+



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NOTE that  +%281-r%5E%28n%2B1%29%29+%2F+%281-r%29+=+%28r%5E%28n%2B1%29-1%29+%2F+%28r-1%29+  (r+%3C%3E+1 of course)





n=1:  LHS:  +1+%2B+r%5E1+=+r%2B1++ 

      RHS:  +%28r%5E%281%2B1%29-1%29%2F%28r-1%29+=+%28r%5E2-1%29%2F%28r-1%29+=+r%2B1+   (ok for n=1)



n=k:  Hypothesis is:  +1%2Br%5E1%2Br%5E2+ +...+ +r%5Ek+ = +%28r%5E%28k%2B1%29-1%29%2F%28r-1%29+





Let n=k+1:  Must show LHS for n=k+1 leads to +%28r%5E%28k%2B2%29-1%29%2F%28r-1%29+:



     LHS:  +1%2Br%5E1%2Br%5E2+ +...+ +r%5Ek+%2B+r%5E%28k%2B1%29+

 

           Apply hypothesis to all but r%5E%28k%2B1%29+ term:

         = +%28r%5E%28k%2B1%29-1%29%2F%28r-1%29+%2B+r%5E%28k%2B1%29+   



           Put everything over r-1:

         = +%28r%5E%28k%2B1%29-1%29%2F%28r-1%29+%2B+%28r%5E%28k%2B1%29%28r-1%29%29%2F%28r-1%29+ 



           Simplifying (showing steps):

         = +%28r%5E%28k%2B1%29-1%29%2F%28r-1%29+%2B+%28r%5E%28k%2B2%29-r%5E%28k%2B1%29%29%2F%28r-1%29+    



         = +%28%28r%5E%28k%2B1%29-1%29+%2B+%28r%5E%28k%2B2%29-r%5E%28k%2B1%29%29%29%2F%28r-1%29+    



         = +%28%28r%5E%28k%2B2%29-1%29%29%2F%28r-1%29+    



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