Which of the following statement is correct?
I. The value of 100² - 99² + 98² - 97² + 96² - 95² + 94² - 93² + . . . . + 2² - 1² is 5050.
II. If 8x + $$\frac{8}{{\text{x}}}$$ = -16 and x < 0, then the value of x¹⁹⁷ + x^-197 is 2.

Solution(By Examveda Team)

$$\text{I}\text{. }\underset{}{\underbracket{{100}^2-{99}^2}}+{98}^2-{97}^2+....+2^2-1^2$$
$$\eqalign{ & = \left( {100 + 99} \right)\left( {100 - 99} \right) + \left( {98 + 97} \right)\left( {98 - 97} \right) + \left( {96 + 95} \right)\left( {96 - 95} \right) + \,.\,.\,.\,.\, + \left( {2 + 1} \right)\left( {2 - 1} \right) \cr & \Rightarrow \frac{{n\left( {n + 1} \right)}}{2} = {\text{ sum of }}n{\text{ terms}} \cr & = \frac{{100 \times 101}}{2} \cr & = 5050 \cr & {\text{Statement I is true}} \cr & {\text{II}}{\text{. }}8x + \frac{8}{x} = - 16 \cr & {\text{Put }}x = - 1 \cr & {\text{Then }}{x^{197}} + {x^{ - 197}} \cr & = {\left( { - 1} \right)^{197}} + {\left( { - 1} \right)^{ - 197}} \cr & = - 1 - 1 \cr & = - 2 \cr & {\text{Statement II is incorrect}} \cr} $$