Question

A=2^65 ; B=2^64+2^63+.........+2^1+2^0 ; C=2^64+2^63+......2^2+2^1 ; then show that 2A=B+C+3

Answer 1

A=2^65 ; B=2^64+2^63+.........+2^1+2^0 ; C=2^64+2^63+......2^2+2^1

Sum of the series of a G.P.

$S_n = \dfrac{a(r^n-1)}{r-1}$

B=2^64+2^63+.........+2^1+2^0

a = 2^0

r = 2^63 / 2^64 = 2

n = 65

$S_n = \dfrac{2^0(2^{65}-1)}{2-1}$

Sn = 2^65 - 1

C=2^64+2^63+.........+2^1

a = 2^1

r = 2^63 / 2^64 = 2

n = 64

$S_n = \dfrac{2^1(2^{64}-1)}{2-1}$

Sn = 2^65 - 2

Now consider,

2A=B+C+3

LHS:

= B+C+3

= (2^65 - 1) + (2^65 - 2) + 3

= 2 × 2^65 - 1 - 2 + 3

= 2 (2^65)

RHS:

= 2A

= 2 (2^65)

LHS = RHS.