Question Number 217178 by hardmath last updated on 04/Mar/25
$$\mathrm{Find}: \\ $$$$\mathrm{100}-\mathrm{99}+\mathrm{98}-\mathrm{97}+\mathrm{96}-\mathrm{95}+…+\mathrm{2}-\mathrm{1}\:=\:? \\ $$
Answered by MathematicalUser2357 last updated on 04/Mar/25
$$=\left(\mathrm{100}+\mathrm{98}+\mathrm{96}+…{choegangcheck}\:{is}\:{cute}…+\mathrm{6}+\mathrm{4}+\mathrm{2}\right)−\left(\mathrm{99}+\mathrm{97}+\mathrm{95}+…{sorry}\:{for}\:{aggro}…+\mathrm{5}+\mathrm{3}+\mathrm{1}\right) \\ $$$$=\mathrm{2}\left(\mathrm{50}+\mathrm{49}+\mathrm{48}+…+\mathrm{3}+\mathrm{2}+\mathrm{1}\right)−\left\{\mathrm{2}\left(\mathrm{50}+\mathrm{49}+\mathrm{48}+…+\mathrm{3}+\mathrm{2}+\mathrm{1}\right)−\mathrm{50}\right\} \\ $$$${let}\:{that}\:{gibberish}\:\left({i}\:{mean},\:\mathrm{50}+\mathrm{49}+\mathrm{48}+…+\mathrm{3}+\mathrm{2}+\mathrm{1}\right)\:{be}\:“{a}'' \\ $$$$=\mathrm{2}{a}−\left(\mathrm{2}{a}−\mathrm{50}\right)=\mathrm{50} \\ $$
Commented by MathematicalUser2357 last updated on 04/Mar/25
$${or}\:{you}\:{just}\:\mathrm{1}+\mathrm{1}+\mathrm{1}+…+\mathrm{1}+\mathrm{1}+\mathrm{1}=\mathrm{50} \\ $$
Answered by efronzo1 last updated on 04/Mar/25
$$\:=\:\underset{\mathrm{n}=\mathrm{1}} {\overset{\mathrm{50}} {\sum}}\left(\mathrm{100}+\mathrm{98}+\mathrm{96}+…+\mathrm{2}\right)− \\ $$$$\:\:\:\:\:\:\underset{\mathrm{n}=\mathrm{1}} {\overset{\mathrm{50}} {\sum}}\left(\mathrm{99}+\mathrm{97}+\mathrm{95}+…+\mathrm{1}\right) \\ $$$$\:=\:\frac{\mathrm{50}}{\mathrm{2}}\left(\mathrm{102}\right)−\frac{\mathrm{50}}{\mathrm{2}}\left(\mathrm{100}\right) \\ $$$$\:\:=\:\frac{\mathrm{50}}{\mathrm{2}}\left(\mathrm{2}\right)=\:\mathrm{50} \\ $$