Question

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

Answer 1

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, ${S_n} = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right]$, 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,${S_n} = \dfrac{{a\left( {1 - {r^n}} \right)}}{{1 - r}}$, 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, $b = \dfrac{{2ac}}{{a + c}}$.