Найдите сумму:
\(\displaystyle 1+2+3+\ldots+30=\)
Для того, чтобы найти сумму \(\displaystyle 1+2+3+\ldots+30{ \small ,}\) перепишем эту же сумму, но в обратном порядке:
\(\displaystyle 30+\ldots+3+2+1{\small .}\)
Сложим их:
| \(\displaystyle +\) | \(\displaystyle S_{30}\) | \(\displaystyle =\) | \(\displaystyle 1\) | \(\displaystyle +\) | \(\displaystyle 2\) | \(\displaystyle +\) | \(\displaystyle 3\) | \(\displaystyle +\ldots+\) | \(\displaystyle 30\) |
| \(\displaystyle S_{30}\) | \(\displaystyle =\) | \(\displaystyle 30\) | \(\displaystyle +\) | \(\displaystyle 29\) | \(\displaystyle +\) | \(\displaystyle 28\) | \(\displaystyle +\ldots+\) | \(\displaystyle 1\) | |
| \(\displaystyle 2\cdot S_{30}\) | \(\displaystyle =\) | \(\displaystyle 31\) | \(\displaystyle +\) | \(\displaystyle 31\) | \(\displaystyle +\) | \(\displaystyle 31\) | \(\displaystyle +\ldots+\) | \(\displaystyle 31\) |
Перепишем получившееся:
\(\displaystyle \begin{aligned} 2\cdot S_{30}&=\underbrace{(1+30)+(2+29)+(3+28)+\ldots+(30+1)}_{30\, раз}=\\&=\underbrace{31+31+31+\ldots+31}_{30\, раз}=(1+30)\cdot 30{\small .}\end{aligned}\)
Таким образом, получили, что
\(\displaystyle 2\cdot S_{30}= (1+30)\cdot 30{\small .} \)
Деля обе части на \(\displaystyle 2{ \small ,} \) получаем:
\(\displaystyle S_{30}=\frac{(1+30)\cdot 30}{2}=465{\small .}\)
То есть
\(\displaystyle \underbrace{\color{blue}{1}+2+3+\ldots+\color{green}{30}}_{30\, членов\, прогрессии}=\frac{(\color{blue}{1}+\color{green}{30})\cdot 30}{2}=465{\small .}\)
Ответ: \(\displaystyle 465{\small .} \)