Question 21

Given $$A =2^{65}$$ and $$B = (2^{64} + 2^{63} + 2^{62} + ... + 2^{0})$$, which of the following is true?

Solution

$$B = (2^{64} + 2^{63} + 2^{62} + ... + 2^{0})$$ is a GP with first term $$2^{64}$$ and common ratio 2 and the number of terms 65.

So its sum can we written as $$B=(2^{64}+2^{63}+2^{62}+...+2^0=\ \frac{2^{65}-1}{2-1}=2^{65}-1=A-1$$

So we get B= A -1 , which means A is larger than B by 1. So option D is the correct choice.

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