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20 Jan 2021, 10:49
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
75%
(hard)
Question Stats:
52% (01:57) correct
48%
(02:06)
wrong
based on 336
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Two integers A and B are such that A = \(2^{65}\) and B = (\(2^{64}+2^{63}+2^{62}+...+2^0\)). Find the relation between A and B.
A. A and B are equal
B. B is greater than A by 1
C. A is greater than B by 1
D. B is \(2^{64}\) greater than A
E. A is \(2^{64}\) greater than B
A. A and B are equal
B. B is greater than A by 1
C. A is greater than B by 1
D. B is \(2^{64}\) greater than A
E. A is \(2^{64}\) greater than B
20 Jan 2021, 11:19
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IMO C
THE SEQUENCE B IS IN GP in which a is 1 and r =2 , n = 65
Sum of n terms in a GP is a(r^n-1) / r-1
When we substitute the values we get B = 2^65 -1
A is given as 2^65
So A is one greater than B
Posted from my mobile device
THE SEQUENCE B IS IN GP in which a is 1 and r =2 , n = 65
Sum of n terms in a GP is a(r^n-1) / r-1
When we substitute the values we get B = 2^65 -1
A is given as 2^65
So A is one greater than B
Posted from my mobile device
20 Jan 2021, 11:36
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DisciplinedPrep
Great question!
In my solution I'll use the fact that \(k + k = 2k\)
So, for example, \(2^3 + 2^3 = 2(2^3) = 2^1(2^3) = 2^4\)
Let's start by examining the following sum: \(1+2^0+2^1+2^2+2^3+2^4\)
Add the first two values to get: \(2+2^1+2^2+2^3+2^4\)
Rewrite as follows: \(2^1+2^1+2^2+2^3+2^4\)
Add the first two values to get: \((2)(2^1)+2^2+2^3+2^4\)
Rewrite as: \(2^2+2^2+2^3+2^4\)
Add the first two values to get: \((2)(2^2)+2^3+2^4\)
Rewrite as: \(2^3+2^3+2^4\)
Add the first two values to get: \((2)(2^3)+2^4\)
Rewrite as: \(2^4+2^4\)
Add the first two values to get: \((2)(2^4)\)
Rewrite as: \(2^5\)
So, \(1+2^0+2^1+2^2+2^3+2^4=2^5\)
If you spot of the pattern, you'll recognize that \(1+2^0+2^1+2^2+2^3+2^4+.....+2^{64}+2^{65}=2^{66}\)
Important: You'll see that the left side of the above equation is 1 GREATER THAN the value of B
You'll also see that the right side of the above equation equals the value of A
So we can rewrite the above equation as follows: \(1+B=A\)
Answer: C
