Question

Between the numbers 2 and 20, 8 arithmetic means are inserted, then their sum is
A. 88
B. 44
C. 176
D. 68

Answer 1

Hint: This problem is regarding arithmetic progression, in an Arithmetic Progression (AP), the whole sequence has a common difference, d. Where the whole AP follows according to this formula: ${a_n} = {a_0} + nd$, where ${a_0}$ is the initial term. $n$is the number of the term in the progression. The common difference is given by the difference between the nth term and the ${(n - 1)^{th}}$term.
Here the sum of the first and the last term is equal to the sum of the second term and the last second term, and so on till the sum of the two middle terms.
$ \Rightarrow {a_0} + {a_n} = {a_1} + {a_{n - 1}} = {a_2} + {a_{n - 2}}....... = {a_{\dfrac{{n - 1}}{2}}} + {a_{\dfrac{n}{2}}}$

Complete step-by-step solution:
Given that there are eight arithmetic means are inserted between the two numbers 2 and 20.
There are two numbers 2 and 20.
Let the 8 arithmetic means be : ${a_1},{a_2},{a_3},{a_4},{a_5},{a_6},{a_7},{a_8}$
We have to find the sum of these 8 arithmetic means.
When these 8 arithmetic means are inserted 2 and 20, it forms an A.P (Arithmetic Progression), as given below:
$ \Rightarrow 2,{a_1},{a_2},{a_3},{a_4},{a_5},{a_6},{a_7},{a_8},20$
The numbers above form an arithmetic progression.
In any arithmetic progression, the sum of the first and the last term is equal to the sum of the second term and the last second term, and so on till the sum of the two middle terms.
Here the first term in the arithmetic progression is = 2
The last term in the arithmetic progression is = 20
Hence the sum of the first and last terms, the sum of the second term and the last second term, and so on, are expressed below:
$ \Rightarrow 2 + 20 = {a_1} + {a_8} = {a_2} + {a_7} = {a_3} + {a_6} = {a_4} + {a_5}$
x$\therefore {a_1} + {a_8} = {a_2} + {a_7} = {a_3} + {a_6} = {a_4} + {a_5} = 22$
The sum of the 8 arithmetic means is given by:
x$ \Rightarrow {a_1} + {a_2} + {a_3} + {a_4} + {a_5} + {a_6} + {a_7} + {a_8}$
Rearranging the terms in such a way that they can be substituted with the obtained values, as given:
$ \Rightarrow \left( {{a_1} + {a_8}} \right) + \left( {{a_2} + {a_7}} \right) + \left( {{a_3} + {a_6}} \right) + \left( {{a_4} + {a_5}} \right)$
$ \Rightarrow 22 + 22 + 22 + 22$
$ \Rightarrow 88$
$\therefore $The sum of the 8 arithmetic means is 88.
The sum of the 8 arithmetic means is 88.

Option A is the correct answer.

Note: Here while solving this problem we have to know that in any arithmetic progression, the arithmetic mean is the ratio of the sum of the terms to the total number of terms in the arithmetic progression as given by:
$ \Rightarrow \dfrac{{{a_1} + {a_2} + {a_3}.......... + {a_{n - 1}} + {a_n}}}{n}$