Question

Arithmetic Sequence? Find the first five terms of the sequence, and determine whether it is arithmetic. If it is arithmetic, find the common difference, and express the $n\text{th}$ term of the sequence in the standard form $a_n=a+\left(n-1\right)d$.

$a_n=1+\frac{n}{2}$

Short answer

The first five terms of the sequence are $\frac{3}{2},2,\frac{5}{2},3,\frac{7}{2}$. The given sequence is an arithmetic sequence with a common difference $\frac{1}{2}$. The $n\text{th}$ term of the sequence is $a_n=\frac{3}{2}+\frac{1}{2}\left(n-1\right)$.

Step-by-step solution

Step 1. Given information.

The given sequence is:

$a_n=1+\frac{n}{2}$

Step 2. Write the concept.

A sequence is called an arithmetic sequence if the difference between consecutive terms is constant.

The $n\text{th}$ term of an arithmetic sequence is:

$a_n=a+\left(n-1\right)d$

Where $a$ is the first term and $d$ is a common difference.

Step 3. Determine the first five terms.

Substitute $n=1,2,3,4,5$in $a_n=1+\frac{n}{2}$to find the first five terms of the sequence.

$\begin{array}{c}{a}_1=1+\frac{1}{2}\\ =\frac{2+1}{2}\\ =\frac{3}{2}\\ a_2=1+\frac{2}{2}\\ =1+1\\ =2\\ a_3=1+\frac{3}{2}\\ =\frac{2+3}{2}\\ =\frac{5}{2}\\ a_4=1+\frac{4}{2}\\ =1+2\\ =3\\ a_5=1+\frac{5}{2}\\ =\frac{2+5}{2}\\ =\frac{7}{2}\end{array}$

The first five terms of the sequence are$\frac{3}{2},2,\frac{5}{2},3,\frac{7}{2}\text{.}$

Step 4. Determine the difference between consecutive terms.

The difference between consecutive terms:

$\begin{array}{c}2-\frac{3}{2}=\frac{4-3}{2}\\ =\frac{1}{2}\\ \frac{5}{2}-2=\frac{5-4}{2}\\ =\frac{1}{2}\\ 3-\frac{5}{2}=\frac{6-5}{2}\\ =\frac{1}{2}\\ \frac{7}{2}-3=\frac{7-6}{2}\\ =\frac{1}{2}\end{array}$

Since the difference between consecutive terms is constant, therefore the given sequence is an arithmetic sequence with the first term $\frac{3}{2}$ and a common difference $\frac{1}{2}$.

Step 5. Determine the nth term.

Substitute $a=\frac{3}{2}$and $d=\frac{1}{2}$in$a_n=a+\left(n-1\right)d$to find the $n\text{th}$

term.

$\begin{array}{c}{a}_n=\left(\frac{3}{2}\right)+\left(n-1\right)\left(\frac{1}{2}\right)\\ a_n=\frac{3}{2}+\frac{1}{2}\left(n-1\right)\end{array}$

Thus, the first five terms of the sequence are $\frac{3}{2},2,\frac{5}{2},3,\frac{7}{2}\text{.}$The

given sequence is an arithmetic sequence with a common

difference$\frac{1}{2}.$The$n\text{th}$ term of the sequence is$a_n=\frac{3}{2}+\frac{1}{2}\left(n-1\right).$