Question

What is the sum of the arithmetic sequence, $2,9,14,19.........$ if there are $38$ terms?

Answer 1

Hint: If we look at this sequence we can see that it increases with a constant value meaning that it is on arithmetic sequences. $9\left( { + 5} \right),14\left( { + 5} \right),19$…..If we add $5$ to $19$ the answer will be $24$, which is the next number in the sequence. If we wanted to work out of $38$ the number in the sequence we would need to work out the $n$ term of this sequence.
Formula used:
The sum of an arithmetic sequence,
${x_n} = a + \left( {n - 1} \right)d$ is given by
$\sum\limits_{n = 0}^{38} {} = \dfrac{n}{2}\left( {2a + \left( {n - 1} \right)d} \right)$
$\text{difference} = {2^{nd}}\text{term} - {1^{st}}\text{term} = {3^{rd}}\text{term} - {2^{nd}}\text{term}$

Complete step-by-step solution:
The given arithmetic sequence is $9,14,19.......$
There are $38$ terms in the arithmetic sequence.
We have to use the formula of the
The sum of an arithmetic sequence,
\[{x_n} = a + \left( {n - 1} \right)d\]
\[\sum\limits_{n = 0}^{38} {} = \dfrac{n}{2}\left( {2a + \left( {n - 1} \right)d} \right)\]
Here $a$ is the first term
$d$ is the difference between each term. There are totally $38$ terms.
Here first term $a$ $ = 9$
Difference $d = 14 - 9 = 19 - 14 = 5$
$d = 5$
Total number of terms $n = 38$
Substitute the values in the equation we have,
$\sum\limits_{n - 0}^{38} {} = \dfrac{n}{2}\left( {2a + \left( {n - 1} \right)d} \right)$
$ = \dfrac{{38}}{2}\left[ {2\left( 9 \right) + \left( {38 - 1} \right)5} \right]$
$ = \dfrac{{38}}{2}\left[ {18 + \left( {37} \right)\left( 5 \right)} \right]$
Add the value, we get,
$ = 19\left( {18 + 185} \right)$
$ = 19 \times 203$
$ = 3857$
Therefore the sum of the arithmetic sequence is $3857$

Note: In the arithmetic sequence it increases with a constant value and has a common difference. Common difference is the proper way and easy way to solve the question.
Additional information:
An Arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is called constant. For instance, the sequence \[5,,7,9,11,13,15,........\] is an arithmetic progression with a common difference of \[2\].
A finite portion of an arithmetic progression is called a finite arithmetic progression and sometimes just called an arithmetic progression. The sum of a finite arithmetic progression is called an arithmetic series.