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What is the sum of all odd numbers between 0 and 100?

Answer
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Hint: We need to find the sum of all odd numbers between 0 and 100. We start to solve the given question by finding the number of odd numbers between 0 and 100 using the arithmetic progression formula an=a+(n1)d . Then, we find the sum of all odd numbers between 0 and 100 using the formula Sn=n2(a+l) to get the desired result.

Complete step by step solution:
We are asked to calculate the sum of all odd numbers between 0 and 100. The given question can be solved using the concept of arithmetic progression.
Odd numbers are the numbers that cannot be divided exactly into pairs. All the odd numbers are not exactly divisible by 2.
Example: 1, 3, 5.
Arithmetic progression is a progression in which the difference between the consecutive terms is a constant.
The formula for the nth term of the arithmetic progression is given as follows,
an=a+(n1)d
Here,
an is the nth term of the sequence
a is the first term of the sequence
n is the number of terms
d is the common difference
We need to find the number of odd numbers between 0 and 100. It is the same as finding the number of odd numbers from 1 to 99.
The sequence of odd numbers between 0 and 100 is given as follows,
1,3,5,7...........99
From the above,
an = 99
a = 1
d = difference between any two consecutive terms = 3 - 1 = 2
Substituting the above values in the formula, we get,
99=1+(n1)2
Moving the term 1 to the other side of the equation, we get,
991=(n1)2
Simplifying the above equation, we get,
98=(n1)2
Dividing the equation by 2 on both sides, we get,
982=(n1)22
Canceling the common factors, we get,
49=(n1)
Moving the term 1 to the other side of the equation, we get,
n=49+1
n=50
Now,
The sum of n terms of an arithmetic progression is given as follows,
Sn=n2(a+l)
Here,
Sn is the sum of n terms
a is the first term
l is the last term
n is the number of terms
The sequence of the sum of odd numbers between 0 and 100 is given as follows,
1+3+5+7+...........+99
From the above,
a = 1
l = 99
n = 50
Substituting the values in the formula, we get,
S50=502(1+99)
Simplifying the above equation, we get,
S50=502(100)
Canceling the common factors, we get,
S50=25×100
S50=2500

The sum of all odd numbers between 0 and 100 is 2500.

Note: The given question can be solved alternatively as follows,
The sum of n consecutive odd numbers is given by the formula,
1+3+5+....+(2n1)=n2
According to the question, we have to find the sum of the sequence
1+3+5+7+...........+99
From the above,
2n1=99
Moving the term 1 to the other side of the equation, we get,
2n=100
Dividing the equation by 2 on both sides, we get,
2n2=1002
n=50
Substituting the value of n in the above formula, we get,
(50)2
2500
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