2條簡單數學題

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2015-03-15 17:51:29 補充：
對，我最初也想這樣做，但從發問者的其他題目看來，他應該未學 sum of geometric series，所以我用了另一個方法。

1. Alternative method :

The binary number is "1111......111" (n digits).

The binary number
= 1×2ⁿ⁻¹ + 1×2ⁿ⁻² + 1×2ⁿ⁻³ + ...... + 1×2² + 1×2 + 1
= 1 + 1×2 + 1×2² + ...... + 1×2ⁿ⁻³ + 1×2ⁿ⁻² + 1×2ⁿ⁻¹

2015-03-15 17:39:52 補充：
It is the sum of a geometric progression with :
first term, a = 1
common ratio, r = 2
number of terms = n

The binary number
= a (rⁿ - 1) / (r - 1)
= 1 × (2ⁿ - 1) / (2 - 1)
= 2ⁿ - 1