Question

If the arithmetic mean of the first n numbers of a series is $\bar{x}$ and the sum of the first (n-1) numbers is k, then which one of the following is the nth number of the series?
(a) $\bar{x}-nk$
(b) $n\bar{x}-k$
(c) $k\bar{x}-n$
(d) $nk\bar{x}$

Answer 1

Hint: We know that the arithmetic mean is the ratio of sum of the values and the total number of values. So, we should first find the sum of n terms using the mentioned formula for arithmetic mean. We can then use the following equation to find the nth term, ${{t}_{n}}={{S}_{n}}-{{S}_{n-1}}$.

Complete step-by-step solution:
We know that the mean or arithmetic mean of n numbers is the ratio of sum of n numbers and the number n itself. We can write this mathematically as
Arithmetic mean of n numbers = $\dfrac{\text{Sum of }n\text{ numbers}}{n}$.
Thus, we can also find the sum of n numbers as,
Sum of n numbers = $n\times $Arithmetic mean.
Here, in this question, the arithmetic mean of n number is given to be $\bar{x}$. So, we can use the above formula to find the sum of first n terms of this series. Thus, we have
Sum of first n terms = $n\bar{x}$
Also, we are given that the sum of first (n – 1) terms of this series is k.
Now, we can easily write
Sum of first n terms = nth term + Sum of first (n – 1) terms.
So, by using the above evaluation, we can write
$n\bar{x}$ = nth term + k.
Now, be rearranging the terms, we get
nth term = $n\bar{x}-k$.
Hence, option (b) is the correct answer.

Note: We must remember that the terms mean, arithmetic mean, expectation, expected value, first moment, etc. represent the same thing and are nothing but the average of the given numbers or dataset. But terms geometric mean and hyperbolic mean are completely different terms.