Chapter 9, Problem 3RE

a.

To determine

To find the common difference.

Answer to Problem 3RE

The common difference is $\frac{3}{2}\cdot$

Explanation of Solution

Given information:

Given sequence is,

  $-1,1/2,2,\frac{7}{2}\mathrm{.........}$

CALCULATION:

Given sequence is,

  $-1,\frac{1}{2},2,\frac{7}{2}\mathrm{.........}$

To find the common difference between the above given sequence first find the difference between each consecutive number.

To find the difference, find the difference between the consecutive numbers.

2^nd term1^st term

  $\frac{1}{2}-\left(-1\right)=\frac{1}{2}+1=\frac{3}{2}$

3^rd term2^nd term

  $1-\frac{1}{2}=\frac{3}{2}$

4^th term3^rd term

  $\frac{7}{2}-2=\frac{3}{2}.$

The difference between each number in the sequence is same then there is the common difference is $\frac{\text{'}3\text{'}}{2}.$

Hence,

The common difference is $\frac{3}{2}\cdot$

b.

To determine

To find the 10^th term.

Answer to Problem 3RE

10^th term is $12.5\cdot$

Explanation of Solution

Given information:

Given sequence is,

  $-1,\frac{1}{2},2,\frac{7}{2}\mathrm{........}$

CALCULATION:

Given sequence is,

  $-1,\frac{1}{2},2,\frac{7}{2}\mathrm{.........}$

To find the 10^th term, first find the $nth$ term of the above given sequence

  $a_n=a_1+\left(n-1\right)d$

[Where, $a_n=nth$ term $n=$ natural number, $d=$ common difference, $a_1=$ 1^st term of the sequence]

  $\begin{array}{l}{a}_n=-1+\left(n-1\right)\frac{3}{2}\left[\text{since, }{a}_1=-1,d=\frac{3}{2}\right]\\ a_{10}=-1+\left(10-1\right)\frac{3}{2}\left[\text{since, }n=10\right]\\ =-1+9\times \frac{3}{2}\\ =-1+\frac{27}{2}\\ =12.5\end{array}$

Hence,

The 10^th term is $12.5\cdot$

c.

To determine

An explicit rule for $nth$ term.

Answer to Problem 3RE

An explicit rule for $nth$ term is $\frac{3}{2}n-\frac{5}{2}.$

Explanation of Solution

Given information:

Given sequence is,

  $-1,\frac{1}{2},2,\frac{7}{2}\mathrm{........}$

CALCULATION:

Given sequence is,

  $-1,\frac{1}{2},2,\frac{7}{2}\mathrm{.........}$

To find the $nth$ term of the above given sequence is defined explicit by,

  $a_n=a_1+d\left(n-1\right)$

[Where, $a_1=$ 1^st term, $d=$ common difference $x=$ natural number $a_n=n-th$ of the above given sequence]

  $\begin{array}{l}{a}_n=-1+\frac{3}{2}\left(n-1\right)\left[\text{since, }{a}_1=-1,d=\frac{3}{2}\right]\\ a_n=\frac{3}{2}n-\frac{5}{2}n\end{array}$

Hence,

An explicit rule for $nth$ term is $\frac{3}{2}n-\frac{5}{2}.$