Chapter 8.4, Problem 126E

To determine

To find: the error in the calculation.

Answer to Problem 126E

In final result for sigma notation value of $r$ must be equal to $0$ instead of $1$ and correct solution is result is $\underset{\_}{{\displaystyle \sum _{r=0}^7\left(\begin{array}{c}7\\ r\end{array}\right)}{\left(x\right)}^{7-r}{\left(5y\right)}^r}$ .

Explanation of Solution

Given information:

The calculation in which error needs to be calculated is as follows

  ${\left(x+5y\right)}^7={\displaystyle \sum _{r=1}^7\left(\begin{array}{c}7\\ r\end{array}\right)}{\left(x\right)}^{7-r}{\left(5y\right)}^r$

Formula used:

  ${\left(x+y\right)}^n={\displaystyle \sum _{r=0}^n\left(\begin{array}{c}n\\ r\end{array}\right)}{\left(x\right)}^{n-r}{\left(y\right)}^r$

Calculation:

The calculation in which error needs to be calculated is as follows

  ${\left(x+5y\right)}^7={\displaystyle \sum _{r=1}^7\left(\begin{array}{c}7\\ r\end{array}\right)}{\left(x\right)}^{7-r}{\left(5y\right)}^r$

Correct calculation

Need to simplify following

  ${\left(x+5y\right)}^7$

Apply formula ${\left(x+y\right)}^n={\displaystyle \sum _{r=0}^n\left(\begin{array}{c}n\\ r\end{array}\right)}{\left(x\right)}^{n-r}{\left(y\right)}^r$

  ${\left(x+5y\right)}^7={\displaystyle \sum _{r=0}^7\left(\begin{array}{c}7\\ r\end{array}\right)}{\left(x\right)}^{7-r}{\left(5y\right)}^r$

Hence on comparing the calculation, error is in applying formula that is in result for sigma notation value of $r$ must be equal to $0$ instead of $1$ and correct solution is result is $\underset{\_}{{\displaystyle \sum _{r=0}^7\left(\begin{array}{c}7\\ r\end{array}\right)}{\left(x\right)}^{7-r}{\left(5y\right)}^r}$ .