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1. 1 + (3 + 1)(3² + 1)(3⁴ + 1)(3⁸ + 1)(3¹⁶ + 1)(3³² + 1) is equal to =?

A. $\frac{3^{64} - 1}{2}$

B. $\frac{3^{64} + 1}{2}$

C. 3⁶⁴ - 1

D. 3⁶⁴ + 1

Solution:

1 + (3 + 1)

(3^{2} + 1)

(3^{4} + 1)

(3^{8} + 1)

(3^{16} + 1)

(3^{32} + 1)

= 1 + \frac{1}{2} [(3 - 1) (3 + 1) (3^{2} + 1) (3^{4} + 1) (3^{8} + 1) (3^{16} + 1) (3^{32} + 1)]

= 1 + \frac{1}{2} [(3^{2} - 1) (3^{2} + 1) (3^{4} + 1) (3^{8} + 1) (3^{16} + 1) (3^{32} + 1)]

= 1 + \frac{1}{2} [(3^{4} - 1) (3^{4} + 1) (3^{8} + 1) (3^{16} + 1) (3^{32} + 1)]

\begin{aligned} = 1 + \frac{1}{2} [(3^{8} - 1) (3^{8} + 1) (3^{16} + 1) (3^{32} + 1)] \\ = 1 + \frac{1}{2} [(3^{16} - 1) (3^{16} + 1) (3^{32} + 1)] \\ = 1 + \frac{1}{2} [(3^{32} - 1) (3^{32} + 1)] \\ = 1 + \frac{1}{2} [(3^{64} + 1)] \\ = \frac{2 + 3^{64} - 1}{2} \\ = \frac{3^{64} + 1}{2} \end{aligned}

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