Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n.6 + 12 + 18 + ... + 6n = 3n(n + 1)
What will be an ideal response?
First we show that the statement is true when n = 1.
For n = 1, we get 6 = 3(1)(1 + 1) = 6.
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 k + 1. That is, we need to show that
So we assume that 
is true and add the next term, 
to both sides of the equation.
6 + 12 + 18 + ... + 6k + 6(k + 1) = 3k(k + 1) + 6(k + 1)
= (k + 1)(3k + 6)
= 3(k + 1)(k + 2)
Condition II is satisfied. As a result, the statement is true for all natural numbers n.
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A. vertex: (1, -4)
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