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The value of (2012)(2013)(2014)(2015)+1 is perfect squae. What is the positive root of this? Please help me how to answer this question without the use of calculator.

Note by Roland Casuga
2 years, 12 months ago

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$$n(n+1)(n+2)(n+3)+1 =n(n+3)(n+2)(n+1)+1 =(n^{2}+3n)(n^{2}+3n+2) +1$$ $$=(n^{2}+3n+1-1)(n^{2}+3n+1+1) +1 =(n^{2}+3n+1)^{2}$$ by set $$n=2012$$ and sqr root both sides you'll get what you want · 2 years, 12 months ago