Given the statement: For every integer n > 2, n^2 – 6n + 1 > 0. Is the text belo

ID: 3010760 • Letter: G

Question

Given the statement:

For every integer n > 2, n^2 – 6n + 1 > 0.

Is the text below a correct proof of the statement? Yes or No

Proof: By induction

Let S[n] be the statement that n^2 – 6n + 1 > 0

We show that if S[k] is true for some integer k > 2 then S[k + 1] is also true.

Suppose that k^2 – 6k + 1 > 0 (Induction Hypothesis)

Then (k + 1)^2 – 6(k + 1) + 1 = k^2 + 2k + 1 – 6k – 6 + 1 = (k^2 – 6k + 1) + (2k – 5)

Both bracketed expressions are positive: the first by the Induction Hypothesis, and the second because k > 2. So S[k + 1] is true.

By PMI, S[n] is true for all n > 2.

we are given the inequality n^2 – 6n + 1 > 0 for all integers n >2

lets plug n =3 in the above inequality and see what happens

(3)^2 - 6(3)+1 >0

-8 > 0 , this is false

for n = 4

(4)^2 - 6(4)+1 >0

-7 > 0 , this is false

for n=5

(5)^2 - 6(5)+1 >0

-4 > 0 , this is false

for n = 6

(6)^2 - 6(6)+1 >0

1 > 0 , this is true

so we could see that the inequality is not true for all integers n > 2

hence the given proof is not valid for n > 2 .

Hence the text is not the correct proof of the statement

For every integer n > 2, n^2 – 6n + 1 > 0

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