Find the average of all odd numbers up to $100$.
A. $30$
B. $50$
C. $4$
D. $10$
Answer
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Hint: Here we want to find the average of all odd numbers up to $100$. For that find the sum of all odd numbers up to $100$ and divide it by the number of odd numbers up to $100$. Try it and you will get the answer.
Complete step-by-step answer:
Odd numbers are the numbers that cannot be divided into two separate groups evenly. These numbers are not completely divided by $2$, which means there is some remainder left after division.
Odd numbers are defined as any number which cannot be divided by two. In other words, several forms $2k+1$, where $k\in Z$ (i.e. integers) are called odd numbers. It should be noted that numbers or sets of integers on a number line can either be odd or even. A few more key points:
An odd number is an integer which is not a multiple of $2$.
If these numbers are divided by $2$, the result or remainder should be a fraction or $1$.
In the number line, $1$ it is the first positive odd number.
The average of odd numbers till a given odd number is a simple concept. You just need to find odd numbers till that number then take their sum and divide by the number.
If an average of odd numbers till $n$ is to be found. Then we will find odd numbers from $1$ to $n$ add then divide it by the number of an odd number.
Average$=$(Sum of Observations)/(No.of Observations)
In case of an odd number of terms with the same difference, the average will be the middle term.
In case of an even number of terms with the same difference, the average will be the average of two middle terms.
The average of first $n$ natural numbers$=\dfrac{(n+1)}{2}$.
The average of squares of first $n$ natural numbers$=\dfrac{(n+1)(2n+1)}{6}$.
The average of cubes of first $n$ natural numbers $=\dfrac{n{{(n+1)}^{2}}}{4}$.
The average of odd numbers from $1$ to $n$\[=\dfrac{\text{Last odd number}+1}{2}\].
The average of even numbers from$1$to $n$$=\dfrac{\text{Last even number}+1}{2}$.
The average of squares of first $n$ odd numbers$=\dfrac{(2n+1)(2n-1)}{3}$.
The average of first $n$ odd numbers = $n$.
So sum of odd number up to $100=1+3+5+.....+97+99$
$=(1+99)+(3+97)+....$Up to $25$ pairs
$=100+100+100+...$$25$ times $=2500$
Average of odd numbers up to $100=$ (sum of the odd numbers up to $100$) / total number of odd numbers.
Total number of odd numbers$=50$
So the average of odd numbers$=\dfrac{2500}{50}=50$.
So an average of odd numbers up to $100$ is $50$.
Here Option(B) is correct.
Note: You should know what odd numbers are. Here in the above problem mistakes are made mainly while doing the sum of odd numbers. Jumbling occurs while simplifying. Take utmost care that no terms are missed.
Complete step-by-step answer:
Odd numbers are the numbers that cannot be divided into two separate groups evenly. These numbers are not completely divided by $2$, which means there is some remainder left after division.
Odd numbers are defined as any number which cannot be divided by two. In other words, several forms $2k+1$, where $k\in Z$ (i.e. integers) are called odd numbers. It should be noted that numbers or sets of integers on a number line can either be odd or even. A few more key points:
An odd number is an integer which is not a multiple of $2$.
If these numbers are divided by $2$, the result or remainder should be a fraction or $1$.
In the number line, $1$ it is the first positive odd number.
The average of odd numbers till a given odd number is a simple concept. You just need to find odd numbers till that number then take their sum and divide by the number.
If an average of odd numbers till $n$ is to be found. Then we will find odd numbers from $1$ to $n$ add then divide it by the number of an odd number.
Average$=$(Sum of Observations)/(No.of Observations)
In case of an odd number of terms with the same difference, the average will be the middle term.
In case of an even number of terms with the same difference, the average will be the average of two middle terms.
The average of first $n$ natural numbers$=\dfrac{(n+1)}{2}$.
The average of squares of first $n$ natural numbers$=\dfrac{(n+1)(2n+1)}{6}$.
The average of cubes of first $n$ natural numbers $=\dfrac{n{{(n+1)}^{2}}}{4}$.
The average of odd numbers from $1$ to $n$\[=\dfrac{\text{Last odd number}+1}{2}\].
The average of even numbers from$1$to $n$$=\dfrac{\text{Last even number}+1}{2}$.
The average of squares of first $n$ odd numbers$=\dfrac{(2n+1)(2n-1)}{3}$.
The average of first $n$ odd numbers = $n$.
So sum of odd number up to $100=1+3+5+.....+97+99$
$=(1+99)+(3+97)+....$Up to $25$ pairs
$=100+100+100+...$$25$ times $=2500$
Average of odd numbers up to $100=$ (sum of the odd numbers up to $100$) / total number of odd numbers.
Total number of odd numbers$=50$
So the average of odd numbers$=\dfrac{2500}{50}=50$.
So an average of odd numbers up to $100$ is $50$.
Here Option(B) is correct.
Note: You should know what odd numbers are. Here in the above problem mistakes are made mainly while doing the sum of odd numbers. Jumbling occurs while simplifying. Take utmost care that no terms are missed.
Last updated date: 24th Sep 2023
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