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Question: How do you find the arithmetic means for the sequence \( - 8, - , - , - , - 24\)...

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

Explanation

Solution

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 aa
For the sequence a1,a2....an{a_1},{a_2}....{a_n} the arithmetic mean an=a1+(n−1)d.{a_n} = {a_1} + \left( {n - 1} \right)d.
Where,
a=a = first term of the sequence.
n=n = number of terms in the sequence.
d=d = difference between each term.
an=nth{a_n} = {n^{th}} term of sequence.
In the sequence a1,a2,a3..{a_1},{a_2},{a_3}.. and difference between two terms is d.d.
Then we can write a2=a1+d{a_2} = {a_1} + d
a3=a2+d{a_3} = {a_2} + d and so on.

Complete step by step solution:
Given sequence is −8,−,−,−,−24 - 8, - , - , - , - 24
Here first term a1=−8{a_1} = - 8 and last term an=−24{a_n} = - 24
The number of terns is the sequence is n=5n = 5
So, we need to find the difference between each terms which is d.d.
So, we have the formula,
an=a1+(n−1)d.{a_n} = {a_1} + \left( {n - 1} \right)d.
a5=a1+(5−1)d.{a_5} = {a_1} + \left( {5 - 1} \right)d.
−24=−8+(5−1)d- 24 = - 8 + \left( {5 - 1} \right)d
−24=−8+4d- 24 = - 8 + 4d
−24+8=4d- 24 + 8 = 4d
−16=4d- 16 = 4d
d=−4d = - 4
The difference between each term is −4 - 4
Now,
a2{a_2} can be written as
a2=a1+d.{a_2} = {a_1} + d.
Were a1=−8{a_1} = - 8
a2=−8−4{a_2} = - 8 - 4
a2=−12{a_2} = - 12
a3{a_3} can be written as,
a3=a2+d{a_3} = {a_2} + d
Here, a2=−12{a_2} = - 12
a3=−12−4{a_3} = - 12 - 4
a3=−16{a_3} = - 16
a4{a_4} can be written as,
a4=a3+d{a_4} = {a_3} + d
Here a3=−16{a_3} = - 16
a4=−16−4{a_4} = - 16 - 4
a4=−20{a_4} = - 20
And the term a5{a_5} is given which is −24 - 24
The arithmetic means for the given sequence which are missing is −12,−16,−20 - 12, - 16, - 20
And sequence can be written as −8,−12,−16,−20,−24 - 8, - 12, - 16, - 20, - 24

Hence the middle term is −16 - 16 which is the mean or the average of the sequence.
∴−8−12−16−20−245=−805=−16\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 8585 and 9595
So, the sequence between 85−9585 - 95
Middle term (mean) =85+952 = \dfrac{{85 + 95}}{2}
=1802= \dfrac{{180}}{2}
=90= 90
a1=85,an=95{a_1} = 85,{a_n} = 95
Here, we have three terms
So, n=3n = 3
We have to find d.d.
So, an=a1+(n−1)d{a_n} = {a_1} + \left( {n - 1} \right)d
95=85+(3−1)d95 = 85 + \left( {3 - 1} \right)d
95=85+2d95 = 85 + 2d
95−85=2d95 - 85 = 2d
10=2d10 = 2d
d=5d = 5

Note: Choose the first term and nth{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. a2=a1+d{a_2} = {a_1} + d