tabita57i

2021-08-10

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 nth term of the sequence in the standard form ${a}_{n}=a+\left(n-1\right)d$
${a}_{n}=1+\frac{n}{2}$

Cristiano Sears

To find:
The sequence's first five terms to determine whether or not it is an arithmetic sequence. If it is arithmetic sequence, find the common difference, and express the nth term of the sequence in the standard form ${a}_{n}=a+\left(n-1\right)d$
Concept used:
In an arithmetic sequence, there is a fixed difference between each succeeding term.
An arithmetic sequence's nth term is determined by,
${a}_{n}=a+\left(n-1\right)d$ .......(1)
Calculation:
Given sequence is,
${a}_{n}=1+\frac{n}{2}$ .......(2)
Substitute 1 for n in equation (2) to calculate the first term of this sequence.
${a}_{1}=1+\frac{1}{2}$
$=\frac{3}{2}$
Substitute 2 for n in equation (2) to calculate the second term of this sequence.
${a}_{2}=1+\frac{2}{2}$
$=1+1$
$=2$
Substitute 3 for n equation (2) to calculate the third term of this sequence.
${a}_{3}=1+\frac{3}{2}$
$=\frac{5}{2}$
Substitute 4 for n equation (2) to calculate the fourth term of this sequence.
${a}_{4}=1+\frac{4}{2}$
$=1+2$
$=3$
Substitute 5 for n equation (2) to calculate the fifth term of this sequence.
${a}_{5}=1+\frac{5}{2}$
$=\frac{7}{2}$
Difference between first and second term can be calculated as,
${a}_{2}-{a}_{1}=2-\frac{3}{2}$
$=\frac{1}{2}$
Difference between second and third term can be calculated as,
${a}_{3}-{a}_{2}=\frac{5}{2}-2$
$=\frac{1}{2}$
Difference between third and fourth term can be calculated as,
${a}_{4}-{a}_{3}=3-\frac{5}{2}$
$=\frac{1}{2}$
Difference between fourth and fifth term can be calculated as,
${a}_{5}-{a}_{4}=\frac{7}{2}-3$
$=\frac{1}{2}$
Since, the difference between the successive terms of the given sequence is constant.
Therefore, the sequence ${a}_{n}=1+\frac{n}{2}$ is not arithmetic sequence.
First term of the sequence ${a}_{n}=1+\frac{n}{2}$ is $a=\frac{3}{2}$
Common difference of the sequence ${a}_{n}=1+\frac{n}{2}$ can be calculated as,
$d={a}_{2}-{a}_{1}$
$=\frac{1}{2}$
Substitute $\frac{3}{2}$ for a and $\frac{1}{2}$ for d in equation (1) to obtain the expression for ${a}_{n}$
${a}_{n}=a+\left(n-1\right)d$
$=\frac{3}{2}+\left(n-1\right)\frac{1}{2}$
Conclusion:
Hence, the sequence ${a}_{n}=1+\frac{n}{2}$ is an arithmetic sequence, the common difference d is $\frac{1}{2}$ and the nth term of the sequence in e standard form is

Jeffrey Jordon