Part 6

Number Theory Level pending

Calculate: $\left \lfloor \frac{1^2 + 1 + 1 }{1\times 2}+\frac{2^2 + 2 + 1}{2\times 3} + \ldots \frac{ n^2 + n + 1 } { n \times (n+1)} + \ldots +\frac{4066273}{4066272} \right \rfloor.$

The answer is 2016.

1 solution

Son Nguyen
Jan 11, 2016

Rewrite the polynomials: $\left [ \frac{3}{1\times 2}+\frac{7}{2\times 3}+\frac{2016^2+2016+1}{2016(2016+1)} \right ]$ We prove that: $\left [ \frac{3}{1\times 2}+\frac{7}{2\times 3}+\frac{n^2+n+1}{n(n+1)} \right ]$ $\frac{3}{1\times 2}=1+\frac{1}{2}$ $\frac{7}{2\times 3}=\frac{1}{2}+\frac{2}{3}$ $\frac{n^2+n+1}{n(n+1)}=n+\frac{n}{n+1}$ $\Rightarrow \sum =\left [ \frac{3}{1\times 2}+\frac{7}{1\times 3}+\frac{n^2+n+1}{n(n+1)} \right ]=n$ Hence n=2016 so the answer is 2016

The answer should be 3 right. There seems to be a mistake in your factorization in your third last sentence.

Pranav Rao - 5 years, 5 months ago

I really like your questions in maths. (Though I can't really solve many of them 😒😒😒) I hope I can learn a thing or two from you. I will wait for reply!

Akhash Raja Raam - 5 years ago

Part 6

The answer is 2016.

1 solution

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