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Question: Find the A.M. of the series 1, 2, 4, 6, 8, 16, ……..,\({2^n}\). A. \(\dfrac{{{2^{n + 1}} - 1}}{{n +...

Find the A.M. of the series 1, 2, 4, 6, 8, 16, ……..,2n{2^n}.
A. 2n+1−1n+1\dfrac{{{2^{n + 1}} - 1}}{{n + 1}}
B. 2n+2−1n\dfrac{{{2^{n + 2}} - 1}}{n}
C. 2n−1n+1\dfrac{{{2^n} - 1}}{{n + 1}}
D. 2n−1n\dfrac{{{2^n} - 1}}{n}

Explanation

Solution

We will first mention the formula of the arithmetic mean and then find the required sum of the given sequence and the number of terms in it to get the required answer.

Complete step-by-step solution:
Let us first write down the formula for the arithmetic mean which is given by the following formula:-
⇒\Rightarrow Arithmetic Mean is given by the Sum of observations divided by the number of observations.
⇒\Rightarrow Arithmetic Mean = Sn\dfrac{S}{n}, where S represents the sum of the observations and n represents the number of observations. ……………….(1)
Since, we are given the sequence 1, 2, 4, ………, 2n{2^n}.
Here if we multiply any term by 2, we will get the succeeding term. Like the first term is 1 and when we multiply it by 2, we get 2.
So, the sequence is a Geometric Progression with the common ratio of 2.
Here, a = 1 and r = 2.
We know that in a G.P., the pth{p^{th}} term is given by ap=arp−1{a_p} = a{r^{p - 1}}
Putting the given and known values, we will get:-
⇒2n=1×2p−1\Rightarrow {2^n} = 1 \times {2^{p - 1}}
⇒n=p−1\Rightarrow n = p - 1
⇒p=n+1\Rightarrow p = n + 1
Hence, there are n + 1 terms in this sequence.
We know that in a G.P., the sum of n terms is given by the formula given by following expression:-
⇒Sn=a(rn−1)(r−1)\Rightarrow {S_n} = \dfrac{{a({r^n} - 1)}}{{(r - 1)}}
Putting the values known to us, we will get:-
⇒Sn+1=1(2n+1−1)(2−1)\Rightarrow {S_{n + 1}} = \dfrac{{1({2^{n + 1}} - 1)}}{{(2 - 1)}}
Simplifying the calculations in the above expression, we will obtain:-
⇒Sn+1=(2n+1−1)\Rightarrow {S_{n + 1}} = ({2^{n + 1}} - 1) …………………..(2)
Now, let us put this in the equation (1):-
⇒\Rightarrow Arithmetic Mean = 2n+1−1n+1\dfrac{{{2^{n + 1}} - 1}}{{n + 1}}

Hence, the correct option is (A).

Note: The students must note that if they take n instead of p while finding the number of terms, it will become confusing as there is n also involved in the last term as well.
Putting that, we will get n + 1 = n, which is definitely absurd.
Therefore, remember to switch it to some other variable to make it right and clear.
The students must commit to memory the following formula:-
⇒\Rightarrow Arithmetic Mean = Sn\dfrac{S}{n}, where S represents the sum of the observations and n represents the number of observations.