Chapter 12.5, Problem 48E

(a)

To determine

To find:whether the equation is true or false explained with answer.

Answer to Problem 48E

The statement is false.

Explanation of Solution

Given:

  

Concept used:

Often mathematical formulae require the addition of many variables Summation or sigma notation is a convenient and simple form of shorthand used to give a concise expression for a sum of the values of a variable.

Let $x_1,x_2,x_3,\dots x_n$ denote a set of n numbers.  $x_1$  is the first number in the set. $x_i$  represents the $ith$ number in the set.

Calculation:

No, it is false in the L.H.S, the term $3^7$ will be repeated whereas it will only occur once in the R.H.S.

Hence, the statement is false.

(b)

To determine

To find:whether the equation is true or false explained with answer.

Answer to Problem 48E

The equation is true.

Explanation of Solution

Given:

  

Concept used:

Often mathematical formulae require the addition of many variables Summation or sigma notation is a convenient and simple form of shorthand used to give a concise expression for a sum of the values of a variable.

Let $x_1,x_2,x_3,\dots x_n$ denote a set of n numbers.  $x_1$  is the first number in the set. $x_i$  represents the $ith$ number in the set.

Calculation:

Left hand side:

  ${\displaystyle \sum _{n=2}^8\left(2n-3\right)}=1+3+5+7+9+11+13.$

Right hand side:

  ${\displaystyle \sum _{n=2}^8\left(2n-3\right)}=1+3+5+7+9+11+13.$

Hence, the equation is true.

(c)

To determine

To find:whether the equation is true or false explained with answer.

Answer to Problem 48E

It is true that the equation is equal.

Explanation of Solution

Given:

  

Concept used:

Often mathematical formulae require the addition of many variables Summation or sigma notation is a convenient and simple form of shorthand used to give a concise expression for a sum of the values of a variable.

Let $x_1,x_2,x_3,\dots x_n$ denote a set of n numbers.  $x_1$  is the first number in the set. $x_i$  represents the $ith$ number in the set.

Calculation:

L.HS:

  $2{\displaystyle \sum _{n=3}^7{n}^2}=2\left(3^2+4^2+5^2+6^2+7^2\right).$

R.H.S:

  $\begin{array}{l}{\displaystyle \sum _{n=3}^{7}2n^2=}{2.3}^2+{2.4}^2+\mathrm{........}+{2.7}^2.\\ \Rightarrow 2\left(3^2+4^2+5^2+6^2+7^2\right).\end{array}$

Hence, its true that the equation is equal.