Chapter 9.2, Problem 36E

Solution

To identify: The sum of any row of Pascal’s triangle is an even integer is true or false.

The sum of any row of Pascal’s triangle is an even integer is true

Given information:

The sum of any row of Pascal’s triangle

Explanation:

Consider the given expression

Use the Binomial Theorem to expand ${\left(a+b\right)}^n={\displaystyle \sum _{r=0}^{n}c{}_r{a}^{n-r}{b}^r}$

When $a=b=1$

  $\begin{array}{c}{\left(1+1\right)}^n={\displaystyle \sum _{r=0}^{50}c{}_r}{\left(1\right)}^{n-r}{\left(1\right)}^r\\ 2^n={\displaystyle \sum _{r=0}^{50}c{}_r}\end{array}$

  $C{}_0+C{}_1+C{}_2+\mathrm{.....}+C{}_n=2^n\mathrm{...}\left(1\right)$

  $nth$ row elements of pascal’s triangle are $C{}_0,C{}_1,C{}_2,\mathrm{.....},C{}_n$

The sum of the elements of $nth$ row elements of pascal’s triangle are which is $2^n$ from (1)

Since for any positive integer n , $2^n$ is even number $C{}_0+C{}_1+C{}_2+\mathrm{.....}+C{}_n$

Hence, the sum of any row of Pascal’s triangle is an even integer is true.