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Prove is true for every natural number by induction.
Let be the proposition that
for the natural number
is true because
This completes the basis step.
INDUCTIVE STEP: For the inductive hypothesis, we assume that is true for an arbitrary natural number That is, we assume that
To carry out the inductive step using this assumption, we must show that when we assume that is true, then is also true. That is, we must show that
assuming the inductive hypothesis Under the assumption of we see that
We have completed the inductive step.
Because we have completed the basis step and the inductive step, by mathematical induction we know that is true for all natural numbers
That is, by principle of mathematical induction
for all natural numbers