Question

Vreau va rog exercitiul 12​

Answer 1

$\it a)~~-1+2-3+4-...-81+82 = 1+1+\underset{de~ \frac{82}{2} = 41~ori}{\underbrace{...}}+1 = \\ = 41\cdot 1 = 41 \\ \\ b)~~1-3+5-7+9-...+121-123 = \\ = (2\cdot 1-1)-(2\cdot1+1)+(2\cdot 2-1)+(2\cdot 2+2)-...+\\ +(2\cdot 66-1)-(2\cdot 66+1) = \\= 2\cdot(1-1+2-2+...+66-66)-1-1-\underset{de~66\cdot 2~ori}{\underbrace{...}}-1= \\ = 2\cdot 0-132\cdot 1 = -132\\ \\ c)~~-2+4-6+8-...-118 =2\cdot (-1+2-3+4-...-59) =$

$\it =2\cdot(-1+2-3+4-...-57+58-59) =\\ = 2\cdot (1+1+\underset{de~\frac{58}{2}~ori}{\underbrace{...}}+1-59) = 2\cdot (\frac{58}{2}-59) = 58-118 = -60\\ \\d)~~1+2-3-4+5+6-7-8+...-159-160 = \\ = \sum\limits_{n=1}^{157}\Big(n+(n+1)-(n+2)-(n+3)\Big) = \sum\limits_{n=1}^{157}(2n-2n+1-5) = \\ = \sum\limits_{n=1}^{157}(-4) = -4\cdot 157 = -628$