[tex]\Bigg[\Big(-\dfrac{1}{2}\Big)\cdot \Big(-\dfrac{1}{2}\Big)^2\cdot \Big(-\dfrac{1}{2}\Big)^3\cdot...\cdot \Big(-\dfrac{1}{2}\Big)^{100}\Bigg]\cdot \Big(-\dfrac{1}{2}\Big)^{-5050} = \\ \\ = \Big(-\dfrac{1}{2}\Big)^{1+2+3+...+100}\cdot \Big(-\dfrac{1}{2}\Big)^{-5050} = \\ \\ = \Big(-\dfrac{1}{2}\Big)^\big{\frac{100\cdot 101}{2}}\cdot \Big(-\dfrac{1}{2}\Big)^{-5050} =[/tex]
[tex]= \Big(-\dfrac{1}{2}\Big)^{5050}\cdot \Big(-\dfrac{1}{2}\Big)^{-5050}= \\ \\ = \Big(-\dfrac{1}{2}\Big)^{5050+(-5050)} = \\ \\ = \Big(-\dfrac{1}{2}\Big)^{0} = \,\boxed{1}[/tex]
