...but I feel like I did something wrong. Also, this was a problem in my Analysis book, hence my posting it here, although it doesn't explicitly deal with analysis.(adsbygoogle = window.adsbygoogle || []).push({});

Prove that root(ab)=(a+b)/2 implies a=b. Assume 0[tex]\leq[/tex]a[tex]\leq[/tex]b. To prove the converse is true was another problem and was easy but anyway here's my work:

Proof.

Assume the contrary; that given root(ab)=(a+b)/2, a[tex]\neq[/tex]b.

By the first multiplicative identity, 2*root(ab)=(a+b).

Squaring both sides: 4ab=a^{2}+2ab+b^{2}

By the first multiplicative identity and algebra, a^{2}-2ab+b^{2}=0.

Factor: (a-b)(a-b)=0.

Since a[tex]\neq[/tex]b by assumptions, a-b[tex]\neq[/tex]0 thus we can divide both sides by a-b: (a-b)=0. This implies a=b, which contradicts the original assumptions. Thus root(ab)=(a+b)/2 implies a=b.

qed

So is this correct? If not, where did I go wrong? I just have this feeling that it's a little...off...somewhere but I don't know how or where. Thanks in advance for the help, this has been bothering me for a bit now.

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# Should be pretty simple proof

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