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Note that if $s_n=\sum_{r=1}^n \frac{1}{r^2} \quad \forall n \in \mathbb{N}-\{1\} \\ s_1=1$ then $\{s_n\}_{n=1}^{\infty}$ is a strictly increasing sequence that which is always less than 2, that is, $n<m \implies s_n<s_m \quad n, m \in \mathbb{N} \\ s_n<2 \quad \forall n \in \mathbb{N}$

Here are the proofs:

The first statement is true as $s_m$ contains more number of positive terms than $s_n$.

To prove the next assertion, note that $\begin{aligned}0&<r(r-1)<r^2 \quad \forall r \in \mathbb{N}-\{1\} \\\implies \frac{1}{r^2} &< \frac{1}{r(r-1)} \quad \forall r \in \mathbb{N}-\{1\} \\\implies s_n&<1+\sum_{r=2}^n \frac{1}{r(r-1)}\\&=1+\sum_{r=2}^n \frac{1}{r-1}-\frac{1}{r}\\&=2-\frac{1}{n}\\\implies s_n&<2 \quad \forall n \in \mathbb{N} \end{aligned}$

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This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

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## Comments

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TopNewestThe answer is 1.

Note that if $s_n=\sum_{r=1}^n \frac{1}{r^2} \quad \forall n \in \mathbb{N}-\{1\} \\ s_1=1$ then $\{s_n\}_{n=1}^{\infty}$ is a strictly increasing sequence that which is always less than 2, that is, $n<m \implies s_n<s_m \quad n, m \in \mathbb{N} \\ s_n<2 \quad \forall n \in \mathbb{N}$

Here are the proofs:

The first statement is true as $s_m$ contains more number of positive terms than $s_n$.

To prove the next assertion, note that $\begin{aligned}0&<r(r-1)<r^2 \quad \forall r \in \mathbb{N}-\{1\} \\\implies \frac{1}{r^2} &< \frac{1}{r(r-1)} \quad \forall r \in \mathbb{N}-\{1\} \\\implies s_n&<1+\sum_{r=2}^n \frac{1}{r(r-1)}\\&=1+\sum_{r=2}^n \frac{1}{r-1}-\frac{1}{r}\\&=2-\frac{1}{n}\\\implies s_n&<2 \quad \forall n \in \mathbb{N} \end{aligned}$

Now, $1=s_1<s_{2016}<2\\\implies \large \boxed{\lfloor s_{2016} \rfloor=1}$

Note:$s_{2016}<\lim_{n \to \infty} s_n = \frac{\pi^2}{6} < 1.645$

Refer here for the proof of this.

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Nice solution..+1.I guessed it anyway

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Try to answer other problems of the set.You are tooooo good in solving problems.

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Will try after jee advanced :)

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I think the answer is 2.

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This is again for NMTC lvl 2 2016

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no it is of 2015

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Oh sorry I meant the previous year IE 2015....I know because I appeared...

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