Which of the following statement is correct?
I. The value of $100^2-99^2+98^2-97^2+96^2-95^2+$ $94^2-93^2+......+2^2-1^2$ is 5050.
II. If $8 x+\frac{8}{x}=-16$ and $x<0$, then the value of $x^{197}+x^{-197}$ is 2.
Answer & explanation
Correct answer: option 1
Checking = $100^2-99^2+98^2-97^2+96^2-95^2+$ $94^2-93^2+......+2^2-1^2$
= (1002 - 992) + (982 - 972) + (962 - 952) + (942 - 932) + ...... + (22 - 12)
= (100 - 99)(100 + 99) + (98 - 97)(98 + 97) + (96 - 95)(96 + 95) + (94 - 93)(94 + 93) + ...... + (2 - 1)(2 + 1)
= 1(100 + 99) + 1(98 + 97) + 1(96 + 95) + 1(94 + 93) + ...... + 1(2 + 1)
= 100 + 99 + 98 + 97 + 96 + 95 + 94 + 93 + ...... + 2 + 1
Now,
We know some n number of consecutive term = n(n + 1)/2
So, [100(100 + 1)]/2
= 50 × 101 = 5050
So, the statement I is correct
Statement II:
If 8x + 8x= -16
= x + 1/x = - 2 [By dividing 8 from both sides]
Now, For x = - 1
x + 1/x = - 2 is satisfying
Now,
x197 + x-197 = x197 + 1/x197
= (- 1)197 + 1/(−1)197
= - 1 – 1 = - 2
Statement II is not correct