Given A = 2⁶⁵ and B = (2⁶⁴ + 2⁶³ + 2⁶² + ..... +2⁰), which of the following is true?

A. B is 2⁶⁴ larger than A

B. A and B are equal

C. B is larger than A by 1

D. A is larger than B by 1

Answer: Option D

Solution(By Examveda Team)

B is in G.P. with a = 2⁰, r = 2, n = 65
$$\eqalign{ & \therefore {S_n} = \frac{{a\left( {{r^n} - 1} \right)}}{{r - 1}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{{2^0}\left( {{2^{65}} - 1} \right)}}{{2 - 1}} \cr & \therefore B = {2^{65}} - 1 \cr & \Rightarrow B = A - 1 \cr & \therefore A\,{\text{is}}\,{\text{larger}}\,{\text{then}}\,B\,{\text{by}}\,1 \cr} $$