Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n.1 + 6 + 62 + ... + 6n - 1 = 

What will be an ideal response?


First, we show that the statement is true when n = 1.
For n = 1, we get 1 (or 6[(1) - 1]) =  =  = 1.
This is a true statement and Condition I is satisfied.

Next, we assume the statement holds for some k. That is,
  is true for some positive integer k.
We need to show that the statement holds for . That is, we need to show that
 
So we assume that  is true and add the next term, 6k, to both sides of the equation.


Condition II is satisfied. As a result, the statement is true for all natural numbers n.

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