Find the sum of all even numbers between 1 to 100.
2550
Finding the Sum of All Even Numbers Between 1 and 100
The problem asks us to find the sum of all even numbers that fall between 1 and 100. The even numbers starting from 1 up to 100 are 2, 4, 6, 8, and so on, until 100.
This sequence of even numbers forms an Arithmetic Progression (AP) because the difference between consecutive terms is constant.
Identifying the Arithmetic Progression (AP)
- The first term (a) of this AP is the first even number, which is 2.
- The last term (l) of this AP is the last even number up to 100, which is 100.
- The common difference (d) between consecutive even numbers is 2 (e.g., 4 - 2 = 2, 6 - 4 = 2).
Calculating the Number of Terms (n)
To find the sum of an AP, we first need to know how many terms are in the sequence. We can use the formula for the n-th term of an AP:
\(\small l = a + (n-1)d\)
Substitute the values we know (l=100, a=2, d=2):
\(\small 100 = 2 + (n-1)2\)
Subtract 2 from both sides:
\(\small 100 - 2 = (n-1)2\)
\(\small 98 = (n-1)2\)
Divide both sides by 2:
\(\small \frac{98}{2} = n-1\)
\(\small 49 = n-1\)
Add 1 to both sides to find n:
\(\small n = 49 + 1\)
\(\small n = 50\)
So, there are 50 even numbers between 1 and 100.
Calculating the Sum of the Even Number Series
Now that we know the number of terms (n=50), the first term (a=2), and the last term (l=100), we can use the formula for the sum of an arithmetic progression:
\(\small S_n = \frac{n}{2}(a+l)\)
Substitute the values:
\(\small S_{50} = \frac{50}{2}(2+100)\)
\(\small S_{50} = 25(102)\)
Multiply 25 by 102:
\(\small S_{50} = 25 \times 102\)
\(\small S_{50} = 25 \times (100 + 2)\)
\(\small S_{50} = (25 \times 100) + (25 \times 2)\)
\(\small S_{50} = 2500 + 50\)
\(\small S_{50} = 2550\)
The sum of all even numbers between 1 and 100 is 2550.
Revision Table: Arithmetic Progression Formulas
| Concept | Formula | Description |
|---|---|---|
| n-th term of AP | \(\small a_n = a + (n-1)d\) | \(a\) = first term, \(d\) = common difference, \(n\) = term number |
| Sum of AP | \(\small S_n = \frac{n}{2}(a+l)\) | \(n\) = number of terms, \(a\) = first term, \(l\) = last term |
| Sum of AP (alternative) | \(\small S_n = \frac{n}{2}(2a + (n-1)d)\) | \(n\) = number of terms, \(a\) = first term, \(d\) = common difference |
Additional Information: Sums of Number Sequences
Besides the sum of an arithmetic progression, there are specific formulas for sums of certain simple sequences:
- Sum of the first n natural numbers: The sum of 1, 2, 3, ..., n is given by \(\small \frac{n(n+1)}{2}\).
- Sum of the first n even numbers: The sum of 2, 4, 6, ..., 2n is given by \(\small n(n+1)\). In our problem, the last term is 100, which is \(2 \times 50\), so n=50. The sum is \(50 \times (50+1) = 50 \times 51 = 2550\). This confirms our result.
- Sum of the first n odd numbers: The sum of 1, 3, 5, ..., (2n-1) is given by \(\small n^2\).
Understanding these basic formulas can help solve many sequence and series problems quickly.
Important Questions from Arithmetic Progressions
The nth term of an A.P is \(\frac{{3 + {\rm{n}}}}{4}\) , then the sum of first 105 terms is
What is the sum of n terms of the series \(\sqrt 2 + \sqrt 8 + \sqrt {18} + \sqrt {32} + \ldots ?\)
The sum of $n$ terms of two arithmetic progressions are in the ratio $(9n + 5) : (5n + 21)$. Find the ratio of their $15^{th}$ terms.
Which of the following disciplines studies human populations mostly with respect to their size, their structure and their development?
What is the sum of all two digit odd numbers?
