Chapter 8.1, Problem 140E

(a).

To determine

To find: first six term of a sequence from the graph.

Answer to Problem 140E

The first six terms of the sequence represented by the graph are $\underset{\_}{1,2,3,4,5,6}$ .

Explanation of Solution

Given information:

The graph which needs to be read to get the first six term of the sequence is as follows.

  

Calculation:

Consider the given graph.

As horizontal axis represents $n$ and vertical axis represents $a_n$ and for all the six points two coordinates are equal so sequence in the graph can be represented as follows

  $a_n=n$

Point $\left(1,1\right)$ means when $n=1$ , $a_1=1$ .

Point $\left(2,2\right)$ means when $n=2$ , $a_2=2$ .

Point $\left(3,3\right)$ means when $n=3$ , $a_3=3$ .

Point $\left(4,4\right)$ means when $n=4$ , $a_4=4$ .

Point $\left(5,5\right)$ means when $n=5$ , $a_5=5$ .

Point $\left(6,6\right)$ means when $n=6$ , $a_6=6$ .

Hence the first six terms of the sequence represented by the graph are $\underset{\_}{1,2,3,4,5,6}$ .

(b).

To determine

To find: the expression of $n^{th}$ term from the graph.

Answer to Problem 140E

Expression representing sequence in the graph is $\underset{\_}{a_n=n}$ .

Explanation of Solution

Given information:

the first six terms of the sequence represented by the graph are $1,2,3,4,5,6$ .

Formula used:

  $a_n=a_1+\left(n-1\right)d$ .

Where $a_n$ is $n^{th}$ term,

  $a_1$ is $1^{st}$ term.

  $d$ is common difference between each consecutive term of sequence..

Calculation:

The first six terms of the sequence are as follows .

  $1,2,3,4,5,6$

First term $a_1=1$ , common difference $d=2-1=3-2=4-3=1$

Substitute $a_1=1$ and $d=1$ in formula $a_n=a_1+\left(n-1\right)d$ .

  $a_n=1+\left(n-1\right)1$

  $a_n=1+n-1$

  $a_n=n$

Hence expression representing sequence in the graph is $\underset{\_}{a_n=n}$ .

(c).

To determine

To find: the sigma notation of Partial sum of the sequence.

Answer to Problem 140E

sigma notation of partial sum of first 50 terms is $\underset{\_}{{\displaystyle \sum _{n=1}^{50}n}}$ .

Explanation of Solution

Given information:

Expression representing sequence in the graph is $a_n=n$ .

Formula used:

  ${\displaystyle \sum _{n=1}^k{a}_n}=1+2+3+\mathrm{4............}n$.

Calculation:

Expression representing sequence in the graph is $a_n=n$ .

Using ${\displaystyle \sum _{n=1}^k{a}_n}=1+2+3+\mathrm{4............}n$

For $n=50$

  $1+2+3+\mathrm{4............50}={\displaystyle \sum _{n=1}^{50}n}$

Hence sigma notation of partial sum of first 50 terms is $\underset{\_}{{\displaystyle \sum _{n=1}^{50}n}}$ .