Question

How do you find the $n{\text{th}}$ term of the sequence $8,12,16,20,24$?

Answer 1

Here we are given the arithmetic progression or we say AP which refers to the sequence in which the common difference is there between any two consecutive terms as here it is $12 - 8 = 16 - 12 = 20 - 16 = 4$.
Here $n{\text{th}}$of the AP is given by ${T_n} = a + \left( {n - 1} \right)d$ and $a$ is first term and $d$ is common difference.

Complete step by step solution:
Here we are given the sequence whose $n{\text{th}}$ term is to be found. Here we have the sequence which is given as $8,12,16,20,24$ and here we know that it is Arithmetic Progression or we say it as AP.
This can be known as here every two consecutive terms of the sequence have the same difference between the numbers like here we have $12 - 8 = 16 - 12 = 20 - 16 = 4$.
So every two consecutive terms have the same common difference which is equal to $4$
Also we have the first term as $8$
We must know that when we need to find the $n{\text{th}}$ term of any sequence which is AP we need to remember the formula of the $n{\text{th}}$ term which is given as:
${T_n} = a + \left( {n - 1} \right)d$
Here $a$ is first term and $d$ is common difference
So we know that first term is $8$ and common difference is $4$
Hence we can say that:
$a = 8 \\\ d = 4 \\\$
So we can put these values in the above formula we will get:
${T_n} = a + \left( {n - 1} \right)d$
${T_n} = 8 + \left( {n - 1} \right)4 = 8 + 4n - 4 \\\ {T_n} = 4 + 4n \\\$

Hence we can say that $n{\text{th term}} = 4 + 4n$

Note:
Here if we are told to find the sum up to the $n$ terms of the sequence we can find it by using the formula which says that:
${S_n} = \dfrac{n}{2}\left( {2a + \left( {n - 1} \right)d} \right)$
Or we can also write ${S_n} = \dfrac{n}{2}\left( {a + l} \right)$ where $l$ is the last term.