Which of the following statement is correct?

I. The value of 100² - 99² + 98² - 97² + 96² - 95² + 942 - 932 + ......  + 222  - 21² is 4840.

II. The value of 

\(\left(k^2+ \frac{1}{k^2} \right) \left(k- \frac{1}{k} \right) \left(k^4+ \frac{1}{k^4} \right) \)\(\left(k+ \frac{1}{k} \right)\left(k^4- \frac{1}{k^4} \right)\) is \(k^{16}- \frac{1}{k^{16}}\).

Concept used:

a² - b² = (a + b) × (a - b)

Tn = a + (n - 1) × d

Where, Tn = last term

a = first term, d = common difference and n = number of terms

Calculation:

if we take the first statement:

1002 - 992 + 982 - 972 + 962 - 952 + 942 - 932 + ......  + 222  - 212 is 4840.

⇒ (100 + 99) × (100 - 99) + (98 + 97) × (98 - 97) ........ + (22 + 21) × (22 - 21)

⇒ 100 + 99 + 98 + 97 ........... + 22 + 21

Now, the series is in A.P

⇒ a + (n - 1) × d = Tn 

⇒ 100 + (n - 1) × (- 1) = 21

⇒ 100 - n + 1 = 21

⇒ - n = (21 - 101) = - 80

⇒ n = 80

Sum of the series : 100 + 99 + 98 + 97 ........... + 22 + 21

⇒ {(100 + 21)/2} × 80

⇒ 60.5 × 80 = 4,840

∴ The first statement is correct.

if we take the second statement:​​​​

\(\left(k^2+ \frac{1}{k^2} \right) \left(k- \frac{1}{k} \right) \left(k^4+ \frac{1}{k^4} \right) \) \(\left(k+ \frac{1}{k} \right)\left(k^4- \frac{1}{k^4} \right)\) is \(k^{16}- \frac{1}{k^{16}}\)

⇒ \(\left(k^2+ \frac{1}{k^2} \right) \left(k^2- \frac{1}{k^2} \right) \left(k^4+ \frac{1}{k^4} \right) \)\(\left(k^4- \frac{1}{k^4} \right)\)

⇒ \(\left(k^4- \frac{1}{k^4} \right) \left(k^4+ \frac{1}{k^4} \right) \)\(\left(k^4- \frac{1}{k^4} \right)\)

⇒ \(\left(k^8- \frac{1}{k^8} \right) \left(k^4- \frac{1}{k^4} \right) \) ≠ \(k^{16}- \frac{1}{k^{16}}\)

∴ The second statement is wrong.

∴ The correct option is 4.