Induction

Mathematical Induction Example 3 --- Geometric Series


Problem: If r is a real number not equal to 1, then for every tex2html_wrap _inline79 ,   tex2html_wrap_inline81 .
Proof:
Basis Step: If n = 0, then LHS = r0 = 1, and RHS = (1 - r) / (1 - r) = 1, since tex2html_wrap_inl ine89 . Hence LHS = RHS.
Induction: Assume that tex2html_wrap_inline81 . -------- Induction Hypothesis
To prove this for n+1,   first try to express LHS for n+1   in terms of LHS for n,   and somehow use the induction hypothesis.
Here let us try
      LHS for n + 1 = tex2html_wrap_inline105 = tex2html_wrap_inline107 .
Using the induction hypothesis, the last expression can be rewritten as tex2html_wrap_inline109 .
Taking the common denominator, it is equal to tex2html_wrap_inline111 ,
which is equal to the RHS for n+1.

Thus LHS = RHS for n+1.

End of Proof.