Sum of a number of terms of $\left\lceil \frac{n}{2} \right\rceil $

Question

The sum of integers from $1$ to $n$ is the following if we ignore the first half of the terms on the left side below:

$ 1+2+...+n\ge \left\lceil \frac{n}{2} \right \rceil + (\left\lceil \frac{n}{2} \right \rceil +1) + ... +n$

$\left\lceil \frac{n}{2} \right \rceil + (\left\lceil \frac{n}{2} \right \rceil +1) + ... +n \ge \left\lceil \frac{n}{2} \right \rceil +...+\left\lceil \frac{n}{2} \right \rceil$

$\left\lceil \frac{n}{2} \right \rceil +...+\left\lceil \frac{n}{2} \right \rceil = (n-\left\lceil \frac{n}{2} \right \rceil+1)*\left\lceil \frac{n}{2} \right \rceil$

$(n-\left\lceil \frac{n}{2} \right \rceil+1)\left\lceil \frac{n}{2} \right \rceil \ge (\frac{n}{2})(\frac{n}{2})$

Could you please explain how we get:

1. $\left\lceil \frac{n}{2} \right \rceil +...+\left\lceil \frac{n}{2} \right \rceil = (n-\left\lceil \frac{n}{2} \right \rceil+1)*\left\lceil \frac{n}{2} \right \rceil$
2. $(n-\left\lceil \frac{n}{2} \right \rceil+1)\left\lceil \frac{n}{2} \right \rceil \ge (\frac{n}{2})(\frac{n}{2})$

Answer 1

1. The number of terms in $a+(a+1)+\dots +b$ is $b-a+1$.
2. Check the two cases $n=2k$ and $n=2k+1$.

Answer 2

$$1+2+\dots+n\ge \left\lceil \frac n2 \right \rceil + \left(\left\lceil \frac n2 \right \rceil +1 \right) + \dots + n$$

Either $n = 2k$ or $n = 2k-1$ for some positive integer, $k$.

In either case, $\left\lceil \dfrac n2 \right\rceil = k \,$ and

\begin{align} \left\lceil \frac n2 \right \rceil + \left(\left\lceil \frac n2 \right \rceil +1 \right) + \dots + n &= k +(k+1) + \dots + n \\ &\le (1 + 2 + \dots +k) + (k +(k+1) + \dots + n) \end{align}

Answer 3

The sum is just a arithmetic series:

$\begin{align*} \sum_{0 \le k \le m} k &= \frac{m (m + 1)}{2} \\ \sum_{\lceil n / 2 \rceil \le k \le n} k &= (n - \lceil n / 2 \rceil + 1) \cdot \lceil n / 2 \rceil + \sum_{0 \le k \le n - \lceil n / 2 \rceil} k \\ &= (n - \lceil n / 2 \rceil + 1) \cdot \lceil n / 2 \rceil + \frac{(n - \lceil n / 2 \rceil) (n - \lceil n / 2 \rceil + 1)}{2} \end{align*}$