Question

The mean of the n observations is \[\overline{X}\] and if the first term is increased by 1 and the second term is increased by 2 and so on, then the new mean is
\[\left( a \right)\overline{X}+n\]
\[\left( b \right)\overline{X}+\dfrac{n}{2}\]
\[\left( c \right)\overline{X}+\dfrac{n+1}{2}\]
\[\left( d \right)\text{None of these}\]

Answer 1

Hint: We will first use the mean formula to calculate the sum of n observations. The mean is given by \[\text{Mean}=\dfrac{\text{Sum of observations}}{\text{Number of observations}}.\] After that we will add 1 to the first term, 2 to the second term and so on and calculate the new sum of the observations. Finally, we will substitute \[\overline{X}\] to get the result. Also, the sum of the n terms is given by \[\dfrac{n\left( n+1 \right)}{2}.\]

Complete step by step answer:
Let us define mean and its formula. Mean is also called the Arithmetic Mean n the average of the numbers also called a central value of a set of numbers. To calculate mean,
1. Add up all the numbers
2. Then divide by how many numbers there are.
The formula is,
\[\text{Mean}=\dfrac{\text{Sum of observations}}{\text{Number of observations}}\]
Then if \[{{x}_{1}},{{x}_{2}}....{{x}_{n}}\] are the terms or the observations, then the mean by using the above formula would be
\[\overline{X}=\dfrac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}+....{{x}_{n}}}{n}.....\left( i \right)\]
Given that 1 is added to the first term. The new first term is
\[\text{First Term}={{x}_{1}}+1\]
And 2 is added to the second term. The new second term is
\[\text{Second Term}={{x}_{2}}+2\]
Similarly, all terms are as below
\[\begin{align}
  & {{x}_{1}}+1 \\
 & {{x}_{2}}+2 \\
 & {{x}_{3}}+3 \\
 & .... \\
 & {{x}_{n}}+n \\
\end{align}\]
The sum of all new observations will be
\[{{x}_{1}}+1+{{x}_{2}}+2+{{x}_{3}}+3.....+{{x}_{n}}+n\]
\[\Rightarrow \left( {{x}_{1}}+{{x}_{2}}+{{x}_{3}}+....+{{x}_{n}} \right)+\left( 1+2+3+....+n \right)\]
So, the sum of new observations is \[\left( {{x}_{1}}+{{x}_{2}}+{{x}_{3}}+....+{{x}_{n}} \right)+\left( 1+2+3+....+n \right).\]
Now, the number of observations is the same ‘n’.
Using the mean of the new observation formula, we have,
\[\text{Mean}\left( \text{new} \right)=\dfrac{\text{Sum of observations}}{\text{Number of observations}}\]
\[\Rightarrow \text{Mean}\left( new \right)=\dfrac{{{x}_{1}}+{{x}_{2}}+.....{{x}_{n}}+\left( 1+2+.....+n \right)}{n}\]
Separating \[\dfrac{1}{n}\] on \[{{x}_{1}}+{{x}_{2}}+....+{{x}_{n}}\] and \[1+2+....+n.\]
\[\Rightarrow \text{Mean}\left( new \right)=\dfrac{{{x}_{1}}+{{x}_{2}}+....+{{x}_{n}}}{n}+\dfrac{1+2+....+n}{n}\]
Substituting the value of \[\overline{X}\] from equation (i) in the above equation, we get,
\[\Rightarrow \text{New Mean}=\overline{X}+\dfrac{\left( 1+2+....+n \right)}{n}\]
Now we know that the sum of the n terms is given by the formula,
\[1+2+3+....+n=\dfrac{n\left( n+1 \right)}{2}\]
Using this in the above equation, we get,
\[\text{New Mean}=\overline{X}+\dfrac{n\left( n+1 \right)}{2\times n}\]
\[\Rightarrow \text{New Mean}=\overline{X}+\left( \dfrac{n+1}{2} \right)\]
Therefore, the new mean is \[\overline{X}+\left( \dfrac{n+1}{2} \right).\]

So, the correct answer is “Option C”.

Note: Another method to solve this question is, after equation (i) write,
\[\overline{X}=\dfrac{{{x}_{1}}+{{x}_{2}}+....+{{x}_{n}}}{n}\]
\[\Rightarrow \overline{X}n={{x}_{1}}+{{x}_{2}}+....+{{x}_{n}}\]
\[\Rightarrow \sum\limits_{i=1}^{n}{{{x}_{i}}=\overline{X}n}\]
where \[\sum\limits_{i=1}^{n}{{{x}_{i}}}\] represents the sum of \[{{x}_{i}}'s.\]
Now, for the new mean,
\[\text{New Mean}=\dfrac{{{x}_{1}}+{{x}_{2}}+....+{{x}_{n}}}{n}+\dfrac{1+2+....+n}{n}\]
\[\Rightarrow \text{New Mean}=\dfrac{\sum\limits_{i=1}^{n}{{{x}_{i}}}}{n}+\dfrac{1+2+....+n}{n}\]
From the above, we have,
\[\Rightarrow \text{New Mean}=\dfrac{\overline{X}n}{n}+\dfrac{n\left( n+1 \right)}{2\times n}\]
\[\Rightarrow \text{New Mean}=\overline{X}+\left( \dfrac{n+1}{2} \right)\]
The answer anyway is the same.