×

# A very strange sequence

There is a sequence defined as follows:

$a_n = \left \lfloor n + \dfrac {1}{2} + \sqrt{n} \right \rfloor$

for all positive $$n$$. Prove that this sequence goes through all non-square positive integers.

Note by Sharky Kesa
2 years, 1 month ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$

## Comments

Sort by:

Top Newest

$$a_{n} = \left\lfloor n + \dfrac{1}{2} + \sqrt{n} \right\rfloor = n + \left\lfloor\dfrac{1}{2} + \sqrt{n} \right\rceil$$
Let $$\sqrt{n} = k + \gamma$$
$$k \in N, 0 \le \gamma < 1$$
$$\therefore a_{n} = n + k + \left\lfloor\dfrac{1}{2} + \gamma \right\rfloor$$
$$a_{n} = n + \lfloor\sqrt{n}\rfloor+ 1 , \gamma \in \left[\dfrac{1}{2},1\right)$$
Or
$$a_{n} = n + \lfloor\sqrt{n}\rfloor, \gamma \in \left[0,\dfrac{1}{2}\right)$$
It suffices to prove that if $$n + \left \lfloor \sqrt{n} \right \rfloor = m^{2}$$ for some integer m, then $$\gamma \in \left[\dfrac{1}{2}, 1\right)$$ .
$$\therefore a_{n} = m^{2} + 1$$ and is not a perfect square.
Similarly if $$n + \left \lfloor \sqrt{n} \right \rfloor + 1 = m^{2}$$ then $$\gamma \in \left[0,\dfrac{1}{2}\right)$$
$$\therefore a_{n} = m^{2}-1$$ and is not a perfect square.

I got stuck at proving the interval $$\gamma$$ belongs in.

- 2 years, 1 month ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...