15Number 15ExplanationGiven that x=12015−22015+32015−42015+⋯−20142015x = \frac{1}{2015} - \frac{2}{2015} + \frac{3}{2015} - \frac{4}{2015} + \dots - \frac{2014}{2015}x=20151−20152+20153−20154+⋯−20152014 The value of xxx that satisfies the equation is...−10072015-\frac{1007}{2015}−20151007−10082015-\frac{1008}{2015}−2015100812015\frac{1}{2015}2015110072015\frac{1007}{2015}2015100710082015\frac{1008}{2015}20151008ExplanationIn addition or subtraction operations, we can use the associative property to simplify the calculation. x=12015−22015+32015−42015+⋯−20142015=(12015−22015)+(32015−42015)+⋯+(20132015−20142015)=(−12015)+(−12015)+⋯+(−12015)\begin{aligned} x &= \frac{1}{2015} - \frac{2}{2015} + \frac{3}{2015} - \frac{4}{2015} + \dots - \frac{2014}{2015} \\ &= \left(\frac{1}{2015} - \frac{2}{2015}\right) + \left(\frac{3}{2015} - \frac{4}{2015}\right) + \dots + \left(\frac{2013}{2015} - \frac{2014}{2015}\right) \\ &= \left(-\frac{1}{2015}\right) + \left(-\frac{1}{2015}\right) + \dots + \left(-\frac{1}{2015}\right) \end{aligned}x=20151−20152+20153−20154+⋯−20152014=(20151−20152)+(20153−20154)+⋯+(20152013−20152014)=(−20151)+(−20151)+⋯+(−20151) There are 20142=1007\frac{2014}{2} = 100722014=1007 groups, so: x=−12015(1007)=−10072015x = -\frac{1}{2015} (1007) = -\frac{1007}{2015}x=−20151(1007)=−20151007
15Number 15ExplanationGiven that x=12015−22015+32015−42015+⋯−20142015x = \frac{1}{2015} - \frac{2}{2015} + \frac{3}{2015} - \frac{4}{2015} + \dots - \frac{2014}{2015}x=20151−20152+20153−20154+⋯−20152014 The value of xxx that satisfies the equation is...−10072015-\frac{1007}{2015}−20151007−10082015-\frac{1008}{2015}−2015100812015\frac{1}{2015}2015110072015\frac{1007}{2015}2015100710082015\frac{1008}{2015}20151008ExplanationIn addition or subtraction operations, we can use the associative property to simplify the calculation. x=12015−22015+32015−42015+⋯−20142015=(12015−22015)+(32015−42015)+⋯+(20132015−20142015)=(−12015)+(−12015)+⋯+(−12015)\begin{aligned} x &= \frac{1}{2015} - \frac{2}{2015} + \frac{3}{2015} - \frac{4}{2015} + \dots - \frac{2014}{2015} \\ &= \left(\frac{1}{2015} - \frac{2}{2015}\right) + \left(\frac{3}{2015} - \frac{4}{2015}\right) + \dots + \left(\frac{2013}{2015} - \frac{2014}{2015}\right) \\ &= \left(-\frac{1}{2015}\right) + \left(-\frac{1}{2015}\right) + \dots + \left(-\frac{1}{2015}\right) \end{aligned}x=20151−20152+20153−20154+⋯−20152014=(20151−20152)+(20153−20154)+⋯+(20152013−20152014)=(−20151)+(−20151)+⋯+(−20151) There are 20142=1007\frac{2014}{2} = 100722014=1007 groups, so: x=−12015(1007)=−10072015x = -\frac{1}{2015} (1007) = -\frac{1007}{2015}x=−20151(1007)=−20151007
