AMC10 2018 A

AMC10 2018 A · Q1

AMC10 2018 A · Q1. It mainly tests Exponents & radicals, Fractions.

What is the value of $(2 + 1)^{-1} + 1^{-1} + 1^{-1} + 1$ ?

$(2 + 1)^{-1} + 1^{-1} + 1^{-1} + 1$ 的值是多少？

(A) $\frac{5}{8}$ $\frac{5}{8}$

(B) $\frac{11}{7}$ $\frac{11}{7}$

(D) $\frac{18}{11}$ $\frac{18}{11}$

(E) $\frac{15}{8}$ $\frac{15}{8}$

Answer

Correct choice: (B)

正确答案：（B）

Solution

Answer (B): Computing inside to outside yields: $\left(\left(\left((2+1)^{-1}+1\right)^{-1}+1\right)^{-1}+1\right)=\left(\left(\left(\frac{4}{3}\right)^{-1}+1\right)^{-1}+1\right)$ $=\left(\left(\frac{7}{4}\right)^{-1}+1\right)$ $=\frac{11}{7}.$ Note: The successive denominators and numerators of numbers obtained from this pattern are the Lucas numbers.

答案（B）：从内到外计算得到： $\left(\left(\left((2+1)^{-1}+1\right)^{-1}+1\right)^{-1}+1\right)=\left(\left(\left(\frac{4}{3}\right)^{-1}+1\right)^{-1}+1\right)$ $=\left(\left(\frac{7}{4}\right)^{-1}+1\right)$ $=\frac{11}{7}。$ 注：由这种模式得到的数的分母与分子依次构成卢卡斯数列（Lucas numbers）。

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