$\"\"$

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The conjecture is $\"\"$ .

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Let $\"\"$ be the statement that $\"\"$ .

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Verify that $\"\"$ is true for $\"\"$ .

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$\"\"$ .

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$

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$\"\"$ is true for $\"\"$ .

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$\"\"$

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Assume that $\"\"$ is true for $\"\"$ .

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$ .

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$\"\"$ is true for positive integer $\"\"$ .

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Show that $\"\"$ must be true.

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$\"\"$

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Add $\"\"$ to each side.

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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The final statement is exactly $\"\"$ , so $\"\"$ is true.

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Because $\"\"$ is true for $\"\"$ and $\"\"$ implies $\"\"$ , $\"\"$ is true for $\"\"$ and so on.

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That is, by the principle of mathematical induction,

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$\"\"$ is true for all positive integers $\"\"$ .

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$\"\"$

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By the principle of mathematical induction,

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$\"\"$ is true for all positive integers $\"\"$ .