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Question: Find the mean of x+3, x+5, x+7, x+9, x+11. A. x+5 B. x+7 C. x+9 D. None of these...

Find the mean of x+3, x+5, x+7, x+9, x+11.
A. x+5
B. x+7
C. x+9
D. None of these

Explanation

Solution

We here have to find the mean of the given observations. For that, we will first see the number of terms given to us of whose we have to find the mean. Then we will use the formula Mean=Sum of observationsnMean=\dfrac{\text{Sum of observations}}{n} where ‘n’ is the total number of observations. Hence, by using this formula, we will find our required answer.

Complete step by step answer:
Here, we have to find the mean of x+3, x+5, x+7, x+9, x+11.
Now, we know that the mean of ‘n’ observations is given as:
Mean=Sum of observationsnMean=\dfrac{\text{Sum of observations}}{n}
Now here, we can see that the number of observations is 5.
Thus here we get, n=5
Hence, we can calculate the median as:
Mean=Sum of observationsn ⇒Mean=(x+3)+(x+5)+(x+7)+(x+9)+(x+11)5 \begin{aligned} & Mean=\dfrac{\text{Sum of observations}}{n} \\\ & \Rightarrow Mean=\dfrac{\left( x+3 \right)+\left( x+5 \right)+\left( x+7 \right)+\left( x+9 \right)+\left( x+11 \right)}{5} \\\ \end{aligned}
Thus, solving this we get:
Mean=(x+3)+(x+5)+(x+7)+(x+9)+(x+11)5Mean=\dfrac{\left( x+3 \right)+\left( x+5 \right)+\left( x+7 \right)+\left( x+9 \right)+\left( x+11 \right)}{5}
Now, adding similar terms together, we get:
⇒Mean=5x+355\Rightarrow Mean=\dfrac{5x+35}{5}
Now, let us separate these terms. Thus we will get:
⇒Mean=5x5+355\Rightarrow Mean=\dfrac{5x}{5}+\dfrac{35}{5}
Now at last dividing the terms by 5 to obtain a non-fractional answer, we get:
∴Mean=x+7\therefore Mean=x+7
Thus, the mean of the given observations is x+7.

Hence, option (B) is the correct option.

Note:
We can here see that the given data is in the form of AP. Thus, we can also find the sum of the series by using the formula for the sum of AP as follows:
Here, a=x+3
d=(x+5)−(x+3)=2d=\left( x+5 \right)-\left( x+3 \right)=2
n=2
Thus, we get the sum ‘S’ as:
S=n2(2a+(n−1)d) ⇒S=52(2(x+3)+(5−1)2) ⇒S=52(2x+6+8) ⇒S=52(2x+14) ∴S=5(x+7)=5x+35 \begin{aligned} & S=\dfrac{n}{2}\left( 2a+\left( n-1 \right)d \right) \\\ & \Rightarrow S=\dfrac{5}{2}\left( 2\left( x+3 \right)+\left( 5-1 \right)2 \right) \\\ & \Rightarrow S=\dfrac{5}{2}\left( 2x+6+8 \right) \\\ & \Rightarrow S=\dfrac{5}{2}\left( 2x+14 \right) \\\ & \therefore S=5\left( x+7 \right)=5x+35 \\\ \end{aligned}