Proving the Result of a Series Inductive Proof

Proof by induction is a powerful tool for proving that a formula for a series works for all terms in the series. In previous examples, we've seen how to prove formulas like $1 + 3 + 5 + \cdots + (2k-1) = k^2$ . Here, we'll work through another example from start to finish.

Recall the two key steps in proof by induction:

Base Case: We show the formula is true for the first term in the series (usually $n=1$ or $n=0$ ). This is like pushing over the first domino.
Inductive Step: We assume the formula is true for some arbitrary term $k$ , and then use this assumption to prove that it must also be true when we add the next term, $k+1$ . This is like showing that if one domino falls, it will knock over the next one.

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