SSC CGL 2022 PYQ — Question 53

The correct answer is option 1 i.e.  $\frac{k^{64}-\frac{1}{k^{64}}}{k+\frac{1}{k}}$

Calculations:

⇒ $(k-\frac{1}{k})(k^2+\frac{1}{k^2})(k^4+\frac{1}{k^4})(k^8+\frac{1}{k^{8}})(k^{16}+\frac{1}{k^{16}})(k^{32}+\frac{1}{k^{32}})$

Multiplying and dividing the above equation by k + 1/k

⇒ $(k-\frac{1}{k})(k^2+\frac{1}{k^2})(k^4+\frac{1}{k^4})(k^8+\frac{1}{k^{8}})(k^{16}+\frac{1}{k^{16}})(k^{32}+\frac{1}{k^{32}}) \times \frac{(k + \frac{1}{k})}{k + \frac{1}{k}}$

Using (a + b)(a - b) = a² - b², we get

⇒ $(k-\frac{1}{k})(k^2+\frac{1}{k^2})(k^4+\frac{1}{k^4})(k^8+\frac{1}{k^{8}})(k^{16}+\frac{1}{k^{16}})(k^{32}+\frac{1}{k^{32}}) \times \frac{1}{k + \frac{1}{k}}$

and using so on this property in the same way we get,

⇒ $\frac{k^{64}-\frac{1}{k^{64}}}{k+\frac{1}{k}}$