Question

How do you find the arithmetic means for the sequence $ - 8, - , - , - , - 24$

Answer 1

Hint: In mathematics and statistics, the arithmetic mean or simply the mean or the average is the sum of a collection of numbers divided by count of numbers in the collection and arithmetic means are the numbers in the sequence which are present. Arithmetic Mean are denoted by the letter $a$
For the sequence ${a_1},{a_2}....{a_n}$ the arithmetic mean ${a_n} = {a_1} + \left( {n - 1} \right)d.$
Where,
$a = $ first term of the sequence.
$n = $ number of terms in the sequence.
$d = $ difference between each term.
${a_n} = {n^{th}}$ term of sequence.
In the sequence ${a_1},{a_2},{a_3}..$ and difference between two terms is $d.$
Then we can write ${a_2} = {a_1} + d$
${a_3} = {a_2} + d$ and so on.

Complete step by step solution:
Given sequence is $ - 8, - , - , - , - 24$
Here first term ${a_1} = - 8$ and last term ${a_n} = - 24$
The number of terns is the sequence is $n = 5$
So, we need to find the difference between each terms which is $d.$
So, we have the formula,
${a_n} = {a_1} + \left( {n - 1} \right)d.$
${a_5} = {a_1} + \left( {5 - 1} \right)d.$
$ - 24 = - 8 + \left( {5 - 1} \right)d$
$ - 24 = - 8 + 4d$
$ - 24 + 8 = 4d$
$ - 16 = 4d$
$d = - 4$
The difference between each term is $ - 4$
Now,
${a_2}$ can be written as
${a_2} = {a_1} + d.$
Were ${a_1} = - 8$
${a_2} = - 8 - 4$
${a_2} = - 12$
${a_3}$ can be written as,
${a_3} = {a_2} + d$
Here, ${a_2} = - 12$
${a_3} = - 12 - 4$
${a_3} = - 16$
${a_4}$ can be written as,
${a_4} = {a_3} + d$
Here ${a_3} = - 16$
${a_4} = - 16 - 4$
${a_4} = - 20$
And the term ${a_5}$ is given which is $ - 24$
The arithmetic means for the given sequence which are missing is $ - 12, - 16, - 20$
And sequence can be written as $ - 8, - 12, - 16, - 20, - 24$

Hence the middle term is $ - 16$ which is the mean or the average of the sequence.
$\therefore \dfrac{{ - 8 - 12 - 16 - 20 - 24}}{5} = \dfrac{{ - 80}}{5} = - 16$

Additional information:
The above question can also be asked in another way.
For example:
What is the arithmetic mean between $85$ and $95$
So, the sequence between $85 - 95$
Middle term (mean) $ = \dfrac{{85 + 95}}{2}$
$ = \dfrac{{180}}{2}$
$ = 90$
${a_1} = 85,{a_n} = 95$
Here, we have three terms
So, $n = 3$
We have to find $d.$
So, ${a_n} = {a_1} + \left( {n - 1} \right)d$
$95 = 85 + \left( {3 - 1} \right)d$
$95 = 85 + 2d$
$95 - 85 = 2d$
$10 = 2d$
$d = 5$

Note: Choose the first term and ${n^{th}}$ term. Carefully because the whole answer depends on these.
Count the number of terms carefully. The second term is addition of the first term and common difference.
i.e. ${a_2} = {a_1} + d$