Prove that the statements in Exercises are true for every positive integer .
To prove: That is true for all n by using mathematical induction.
Considering the given statement is for all integers n,
a. if is true for
b. if the truth of always implies that is true, then the statement is true for all positive integers .
For each positive integer , let be the statement
Here, is the last term on the left side.
When , there is only one term, and no addition is actually performed.
When , the value of the left side is
And the value of right side is
Thus is true.
Assume now that is true. That is, assume that the equation
With this assumption made, prove that is true. Add to both sides of the assumed equality, the following result is obtained
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