Question

The mean of 30 consecutive natural
numbers is x and the mean of the odd
numbers in these 30 natural numbers is y.
If the smallest number is an odd number,
find the value of x - y.​

Answer 1

$\large\text{\underline{Question}}$

The mean of 30 consecutive natural number is $x$ and the mean of the odd numbers in these 30 natural numbers is $y$.

If the smallest number is an odd number, find the value of $x-y$.

$\large\text{\underline{Let's begin.}}$

$\hookrightarrow\text{Sum of the sequence.}$

$\large\boxed{S_{n}=\dfrac{n(a+l)}{2}}$

where,

• $a$ is the first term
• $l$ is the last term
• $n$ is the number of terms

$\hookrightarrow\text{Arithmetic mean.}$

$\boxed{\text{(Arithmetic mean)}=\dfrac{\text{Sum of all terms}}{\text{Number of terms}}}$

$\large\text{\underline{Solution}}$

Let's calculate the sum of 30 consecutive natural numbers.

This sequence is an arithmetic progression. To avoid confusion, I will be using $S_{n_{1}}$ as the sum of natural numbers, and $S_{n_{2}}$ as the sum of odd numbers.

Let the smallest number be $a$.  And let the sum of the sequence be $S_{n_{1}}$.

$S_{n_{1}}=a+(a+1)+(a+2)+\cdots+(a+29)$

$\implies S_{n_{1}}=\dfrac{30(2a+29)}{2}$

Now let's calculate the sum of the odd numbers.

This is also an arithmetic progression. Let the sum of the sequence be $S_{n_{2}}$ this time. Since the smallest number $a$ is an odd number, we get,

$S_{n_{2}}=a+(a+2)+(a+4)+\cdots+(a+28)$

$\implies S_{n_{2}}=\dfrac{15(2a+28)}{2}$

Now we know the sum of consecutive numbers. Now it's time to find the mean of each sum.

$\hookrightarrow\text{Arithmetic mean.}$

$\boxed{\text{(Arithmetic mean)}=\dfrac{\text{Sum of all terms}}{\text{Number of terms}}}$

$\hookrightarrow\text{Mean of consecutive natural numbers.}$

$\implies\dfrac{S_{n_{1}}}{30}=a+14.5$

$\hookrightarrow\text{Mean of consecutive odd numbers.}$

$\implies\dfrac{S_{n_{2}}}{15}=a+14$

$\large\text{\underline{Conclusion}}$

Hence, $x-y=\dfrac{1}{2}$.

$\large\text{\underline{Learn more.}}$

$\hookrightarrow\text{Another form of the arithmetic series.}$

$\large\boxed{S_{n}=\dfrac{n\{2a+(n-1)d\}}{2}}$

where $l$ is substituted by $a+(n-1)d$.

Answer 2

Given : The mean of 30 consecutive natural numbers is x and the mean of the odd numbers in these 30 natural numbers is y.

The smallest number is an odd number,

To find :  the value of x - y.​

Solution:

The smallest number is an odd number,

Assume 30 natural numbers are

2k+1 , 2k + 2 , ...   , 2k + 29 , 2k + 30

15 odd natural numbers in these are

2k + 1 , 2k + 3 , .... , 2k + 29  

where k is non negative integer

Sₙ = (n/2) (first term + nth Term)  

Mean =  ( first Term + nth term)/2

Mean of 30 natural numbers

x= (2k +1  + 2k + 30)/2

x = 2k + 15.5

Mean of 15 odd natural numbers

y= (2k +1  + 2k + 29)/2

y = 2k + 15

x  - y  = 2k + 15.5 - (2k + 15)

=> x - y = 0.5

the value of x - y is 0.5

Learn More:

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