Chapter 9.2, Problem 41E

Solution

a.

To calculate: The first 10 triangular numbers.

Triangular Numbers of the form $1+2+\dots \dots +n$ are triangular numbers because they count numbers in triangular arrays.

  

The 10 terms of the sequence are $T_n=1,3,6,10,15,21,28,36,45,55$

Given information:

The sequence is $1+2+\dots \dots +n$

  

Calculation:

Consider the given triangular numbers

  $1+3+6+\mathrm{...}+n$

The first triangular number is $T=1$ and other triangular number is $T=1,3,6,10\text{and}\text{15}$

The 1st to 10th number.

  $\begin{array}{c}{T}_1=1\\ T_2=3\\ T_3=6\\ T_4=10\end{array}$

The relation is

  $\begin{array}{l}{T}_1=1\\ T_2=2+1\\ T_3=3+3\\ T_4=6+4\end{array}$

So, the nth term will be

  $T_n=T_{n-1}+n$

Hence the 10 terms of the sequence are $T_n=1,3,6,10,15,21,28,36,45,55$

b.

To identify: The triangular numbers appear in Pascal’s triangle?

The sequence goes diagonally marked in pascal triangle.

Given information:

The given sequence is $1,3,6,10,15,21,\mathrm{...}$

Explanation:

Consider the pascal’s triangle as below:

  

Now the sequence $1,3,6,10,15,21,\mathrm{...}$ goes as marked below.

So, the sequence goes diagonally marked in pascal triangle.

c.

To identify: The nth triangular number can be written as $\frac{n\left(n+1\right)}{2}$ .

  

The triangular number will be expressed as $\frac{n\left(n+1\right)}{2}$

Given information:

The given diagram is

  

Explanation:

Consider the rectangular diagram given in the book on age number 716

Here, the rectangle is taken in such a way that each half of it (white or black) is similar to a triangle

Count the dots of the rectangle

The number of horizontal dot of one side is

‘6’

The number of vertical dot is

‘5’

So, if the number of vertical dot will $n$ be then horizontal dot $n+1$

So, the area of the rectangle

  $n\left(n+1\right)$

Each triangle is half of the rectangle.

So, the area is $\frac{n\left(n+1\right)}{2}$

As this is equivalent to triangular numbers

Hence, the triangular number will be expressed as $\frac{n\left(n+1\right)}{2}$

d.

To identify: The formula in part (c) as a binomial coefficient.

The binomial coefficient is $\left(\begin{array}{l}n+1\\ 2\end{array}\right)$ .

Given information:

The given sequence is $\frac{n\left(n+1\right)}{2}$

Explanation:

Consider the given sequence.

Rewrite the given sequence as:

  $\begin{array}{c}\frac{n\left(n+1\right)}{2}=\frac{n\left(n+1\right)}{2}\times \frac{\left(n-1\right)\left(n-2\right)\cdots 1}{\left(n-1\right)\left(n-2\right)\cdots 1}\\ =\frac{n\left(n+1\right)\left(n-1\right)\left(n-2\right)\cdots 1}{2\left(n-1\right)\left(n-2\right)\cdots 1}\\ =\frac{n!}{2\left(n-1\right)!}\\ =\frac{n!}{2!\left(n-1\right)!}\\ =\left(\begin{array}{l}n+1\\ 2\end{array}\right)\end{array}$

The binomial coefficient is $\left(\begin{array}{l}n+1\\ 2\end{array}\right)$ .