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I am working on a problem, assuming high school math knowledge.

Let ${a_n}$ be the sequence defined by $$a_n=\left[\frac{n^2+8n+10}{n+9}\right]\,,$$ where $[x]$ denotes the largest integer which does not exceed $x$. Find the value of $ \sum\limits_{n=1}^{30} a_n$.

I honestly do not understand the text in bold. The answer provided is $445$. Could you please explain what I should be looking at here?

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  • $\begingroup$ E.g. $[\pi] = 3$ while $[-\pi ] = -4$. $\endgroup$
    – xbh
    Commented Jun 19, 2019 at 6:57

3 Answers 3

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Hint \begin{align*} \frac{n^2+8n+10}{n+9}&=\frac{n^2+9n-n-9+19}{n+9}\\ &=n-1+\frac{19}{n+9}\\ \therefore \left\lfloor\frac{n^2+8n+10}{n+9}\right\rfloor&=n-1+\left\lfloor \frac{19}{n+9}\right\rfloor \end{align*} As $n$ varies from $1$ to $30$, we will have \begin{align*} \left\lfloor\frac{19}{\color{red}{1}+9}\right\rfloor&=\lfloor 1.9\rfloor=1\\ \left\lfloor\frac{19}{\color{red}{2}+9}\right\rfloor&=\lfloor 1.7\rfloor=1\\ \vdots & = \vdots\\ \left\lfloor\frac{19}{\color{red}{11}+9}\right\rfloor&=\lfloor 0.95\rfloor=0\\ \vdots & = \vdots\\ \end{align*} Hopefully you can take it from here.

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$[x]$ is the largest integer which does not exceed $x$.

For example, $[12.4]=12$, $[10.995]=10$, $[7]=7$, $[-2.3]=-3$.

For this problem $\displaystyle a_n=\left[\frac{(n-1)(n+9)+19}{n+9}\right]=n-1+\left[\frac{19}{n+9}\right]$.

For $n=1,2,\dots, 10$, $a_n=n$.

For $n=11,12,\dots, 30$, $a_n=n-1$.

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$$\sum\limits_{n=1}^{30} a_n =\sum\limits_{n=1}^{30}(n-1)+ \sum\limits_{n=1}^{30} [\frac{19}{n+9}]=$$

$$ (1+2+3+...+29) + \sum _1^{10} (1) =435+10=445$$

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