Question

If A = 2⁶⁵ and B = 2⁶⁴ + 2⁶³ + 2⁶² …. + 2⁰ identify of the following is true?

Options

• B is 2⁶⁴ more than A

• A and B are equal

• B is larger than A by 1

• A is larger than B by 1

Answer 1

A is larger than B by 1

Explanation;

Hint:

A = 2⁶⁵

B = 2⁶⁴ + 2⁶³ + 2⁶² + … + 20

B = 2⁰ + 2¹ + 2² + … + 264

G.P = 1 + 2¹ + 2² + … + 2⁶⁴ it is a G.P

Here a = 1, r = 2, n = 65

∴ Sum of the G.P = S₆₅

= ${\displaystyle \frac{\text{a}({\text{r}}^{\text{n}}-1)}{\text{r}-1}}$

= ${\displaystyle \frac{1(2^{65}-1)}{2-1}}$

= 265 – 1

A = 2⁶⁵, B = 2⁶⁵ – 1

∴ B is smaller.

A is larger than B by 1.