Question

How do you determine whether each sequence is an arithmetic sequence: 4, 9, 14, 19...?

Answer 1

Hint: In this question we have to find whether the given sequence is in arithmetic progression, this can be done by finding the common difference , we will use the common difference formula which is given by, Common difference $\left(d\right)$ is given by $a_{n+1}-a_n$. , so if all the common differences are equal then we can say that they are in arithmetic progression.

Complete step by step solution:
An arithmetic progression is a sequence where the differences between every two consecutive terms are the same. An arithmetic progression is a sequence where each term, except the first term, is obtained by adding a fixed number to its previous term.
Common difference $\left(d\right)$is given by$a_{n+1}-a_n$.
The sequence is 4, 9, 14, 19……
So, here first term is 4, second term is 9,
Now common difference $=9-4=5$,
And here third term is 14, and second term is 9,
Now common difference $=14-9=5$,
And here third term is 14, and fourth term is 19,
Now common difference $=19-14=5$,
So, they all have the same common difference, which is 5, so, the given sequence is in arithmetic progression.

$\therefore$ The given sequence is in arithmetic progression as they their common difference is same.

Note: There are 3 types of series i.e., Arithmetic series, Geometric series and Harmonic series, here are some useful formulas related to the above series:
Sum of the $n$ terms in A. P is given by, $S_n={\displaystyle \frac{n}{2}}\left[2a+\left(n-1\right)d\right]$, where $n$ is common difference, $a$ is the first term.
The $n^{th}$ term In A.P is given by $T_n=a+\left(n-1\right)d$,
Sum of the $n$ terms in GP is given by,$S_n={\displaystyle \frac{a\left(1-r^n\right)}{1-r}}$, where $r$ is common ratio, $a$ is the first term.
The $n^{th}$ term In A.P is given by $T_n=a{r}^{n-1}$,
If a, b, c are in HP, then b is the harmonic mean between a and c.
In this case, $b={\displaystyle \frac{2ac}{a+c}}$.