Chapter 8.1, Problem 59E

To determine

Whether each of the statement is true. If so provide a proof. If not provide a counterexample.

Answer to Problem 59E

  $aandc$ are true.

  $bandd$ are false.

Explanation of Solution

Given:

  

  $(a){\displaystyle \sum _{i=1}^{n}c{a}_i=c}{\displaystyle \sum _{i=1}^n{a}_i}$

The statement is true.

  $\begin{array}{l}(a){\displaystyle \sum _{i=1}^{n}c{a}_i=c{a}_1}+c{a}_2+c{a}_3+\mathrm{....}+c{a}_n\\ =c(a_1+a_2+\mathrm{...}{a}_n)\\ =c{\displaystyle \sum _{i=1}^n{a}_i}\end{array}$

  $(b){\displaystyle \sum _{i=1}^n(a_i+b_i)}={\displaystyle \sum _{i=1}^n{a}_i}+{\displaystyle \sum _{i=1}^n{b}_i}$

The stamen is true.

  $\begin{array}{l}{\displaystyle \sum _{i=1}^n(a_i+b_i)}=(a_1+b_1)+(a_2+b_2)+\mathrm{...}+(a_n+b_n)\\ =(a_1+a_2+\mathrm{...}+a_n)+(b_1+b_2+\mathrm{...}+b_n)\\ ={\displaystyle \sum _{i=1}^n{a}_i}+{\displaystyle \sum _{i=1}^n{b}_i}\end{array}$

  $(c){\displaystyle \sum _{i=1}^n(a_i)(b_i)}=\left({\displaystyle \sum _{i=1}^n(a_i)}\right)\left({\displaystyle \sum _{i=1}^n(b_i)}\right)$

The statement is false

  $\begin{array}{l}{\displaystyle \sum _{i=1}^2(a_i)(b_i)}=(1)(1)+(2)(2)\\ =1+4\\ =5\\ =\left({\displaystyle \sum _{i=1}^2(a_i)}\right)\left({\displaystyle \sum _{i=1}^2(b_i)}\right)=(1+2)+(1+2)9=\end{array}$

  $(d){\displaystyle \sum _{i=1}^n{(a_i)}^c}={\left({\displaystyle \sum _{i=1}^n(a_i)}\right)}^c$

The statement is false.

  $\begin{array}{l}{\displaystyle \sum _{i=1}^2{(i)}^2=5}\\ ={\left({\displaystyle \sum _{i=1}^2(i)}\right)}^2={(}^1=9\end{array}$