For Each Integer $n \geq 3$ , Let $P ( n )$

question 8

Essay

For each integer $n \geq 3$ , let $P ( n )$ be the equation
$3+4+5+\cdots+n=\frac{(n-2)(n+3)}{2} \cdot \leftarrow P(n)$
(Recall that by definition $3 + 4 + 5 + \cdots + n = \sum _ { i = 3 } ^ { n } i _ {. }$ )
(a) Is $P ( 3 )$ true? Justify your answer.
(b) In the inductive step of a proof that $P ( n )$ is true for all integers $n \geq 3$ , we suppose $P ( k )$ is true (this is the inductive hypothesis), and then we show that $P ( k + 1 )$ is true. Fill in the blanks below to write what we suppose and what we must show for this particular equation.
Proof that for all integers $k \geq 3$ , if $P ( k )$ is true then $P ( k + 1 )$ is true:
Let $k$ be any integer that is greater than or equal to 3 , and suppose that___ We must show that________
(c) Finish the proof started in (b) above.

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Definitions:

Clarity

The quality of being clear, understandable, and free from ambiguity in expression or thought.

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