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Suppose $$n^{2} = n + 1, n \geq 2$$

Prove using induction.

Inductionstart: $$n_{0} = 2$$ Therefore 4 > 3 and the induction start holds.

Inductionstep: $$(n + 1)^{2} = n + 2$$ $$n^{2} + 2n + 2 > n + 2$$

Question:

Is it okay to just drop the $n^{2}$ and compare 2n + 2 > n + 2 since this inequality clearly holds true.

Sorry if this is an obvious question, but looking forward to any help and explanations!

J3ck_Budl7y
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    Your ideas are in the right direction, but if you want to formally prove this, make sure to explicitly say things like “assume the proposition is true for $n=k$; then I will show it is true for $n=k+1$.” Also, I think what you’ll end up comparing is $n^2+2n+1$ and $n+2$. – Philip Speegle Sep 06 '22 at 04:16
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    Thanks, I will write it more formally just wanted to see if it would be considered proved if I approached it in this way – J3ck_Budl7y Sep 06 '22 at 04:21
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    Since $n>=2$, $n^2$ must be greater than or equal to $4$. You can assume $n^2>n+1$ then, so since $2n+2>1$ for $n>=2$, the inequality holds true. – ArthD21 Sep 06 '22 at 04:23
  • This is definitely a more approachable and easy way to prove this inequality @MathAndPhysics, thank you! Nevertheless, the question demanded a prove by induction. Would it still be considered a prove by I induction if I used your approach? – J3ck_Budl7y Sep 06 '22 at 04:31

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What's better is to subtract $n+2$ from both sides to get $n^2+n>0$, which clearly holds.

Kamal Saleh
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