Chapter 10.2, Problem 85STP

To determine

Find the correct option for the expression $1+\sqrt{2}+\sqrt[3]{3}$

Answer to Problem 85STP

Option A is correct choice.

Explanation of Solution

Given:

The expression $1+\sqrt{2}+\sqrt[3]{3}$ is equivalent to $\text{A}:{\displaystyle \sum _{k=1}^3{k}^{\frac{1}{k}}};\text{B}:{\displaystyle \sum _{k=1}^3{k}^k};\text{C}:{\displaystyle \sum _{k=1}^3{k}^{-k}};\text{D}:{\displaystyle \sum _{k=1}^3\sqrt{k}}$

Concept Used:

Let we check each options given by substituting k = 1, 2, 3

Calculation:

Let we check each options given by substituting k = 1, 2, 3

  $\text{A}:{\displaystyle \sum _{k=1}^3{k}^{\frac{1}{k}}}[\text{Substitute}k=1,2,3\text{and}\text{check}]$

  $\text{A}:{\displaystyle \sum _{k=1}^3{k}^{\frac{1}{k}}}\Rightarrow 1^{\frac{1}{1}}=1(k=1);2^{\frac{1}{2}}=\sqrt{2}(k=2);3^{\frac{1}{3}}=\sqrt[3]{3}(k=3);$ Correct Option

  $\text{B}:{\displaystyle \sum _{k=1}^3{k}^k}\Rightarrow 1^1=1(k=1);2^2=4(k=2);3^3=9(k=3)\Rightarrow$ Discarded

  $\text{C}:{\displaystyle \sum _{k=1}^3{k}^{-k}}\text{This}\text{option}\text{is}\text{not}\text{possible}\text{,}\text{cause}\text{there}\text{in}\text{no}\text{negative}\text{exponent}$

  $\text{D}:{\displaystyle \sum _{k=1}^3\sqrt{k}}\Rightarrow \sqrt{1}=1(k=1);\sqrt{2}(k=2);\sqrt{3}(k=3)\Rightarrow$ Discarded

Thus, Correct option is A