Chapter 12.6, Problem 1CFU

(a)

To determine

To find:the sum of the numbers in each row of pascals triangle for $n=0\text{ to }5.$

Answer to Problem 1CFU

In generals it can find the sum of the number of $n^{th}\text{ row=}{2}^n$ .

Explanation of Solution

Given:

  $n=0ton=5$ .

Concept used:

Pascal triangle diagrams in the form of pyramid and the general formula for the sum of value of the row is:

  $2^n$ .

  $1st$ row is considered as $0th$ row.

Example:

  $1st$ row $2^0=1.$

Calculation:

The sum of the numbers of each row of pascals triangle for $n=0ton=5.$

Pascals triangle for $n=0ton=5$ is:

  $\begin{array}{l}1\\ 11\\ 121\\ 1331\\ 14641\\ 15101051\end{array}$

From the above Pascals triangle,

Sum of the numbers of $0\text{th}$ row $1\text{ or }{2}^0$ .

Sum of the numbers of first row $2\text{ or }{2}^1$ .

Sum of the numbers of second row $4\text{ or }{2}^2$ .

Sum of the numbers of third row $8\text{ or }{2}^3$ .

Sum of the numbers of fourth row $16\text{ or }{2}^4$ .

Sum of the numbers of fifth row $\text{32 or }{2}^5$ .

Hence, in generals it can find the sum of the number of $n^{th}\text{ row=}{2}^n$ .

(b)

To determine

To find:the expression to represent the sum of numbers in the nth row of pascals triangle.

Answer to Problem 1CFU

Sum of the numbers in the $nth$ row is $2^n$ .

Explanation of Solution

Given:

  $nth$ row.

Concept used:

Pascal triangle diagrams in the form of pyramid and the general formula for the sum of value of the row is:

  $2^n$ .

  $1st$ row is considered as $0th$ row.

Example:

  $1st$ row $2^0=1.$

Calculation:

From the part $\left(a\right)$ ,sum of the terms in the $nth$ row, is $2^n$ .

Therefore, the expression to represent the sum of the numbers in the $nth$ row is $2^n$ .

Hence, sum of the numbers in the $nth$ row is $2^n$ .