Sum in Math | Definition, Examples & Formulas
Last Updated :
06 Aug, 2025
In mathematics, the term "sum" refers to the result obtained by adding two or more numbers or quantities together. It's a fundamental arithmetic operation used to find the total of multiple values. For example, when you add 5 and 7, the sum is 12.
In this article, we will learn the concept of Sum in mathematics, along with its definition, some examples, and properties of Sum.
Sum Meaning
The sum is the fundamental arithmetic operation that combines two or more quantities by adding them together to find their total. It can be used to add numbers as well as variables. Also, the sum operation is not limited to just numerical values; it can also be applied to algebraic expressions and mathematical variables.
For example, if Ram buys 20 Rs worth of bananas, 40 Rs worth of apples, and 50 Rs worth of oranges, then the total can be found by using the sum operation between 20, 40, and 50
The term "sum" refers to the result obtained by adding two or more numbers or quantities together.
It represents the total value obtained when multiple values are combined through addition.
Sum Notation | Symbol for Sum (+)
The symbol for Sum is +. This is used when we add the numbers together. For a list of numbers in a sequence, we use the uppercase Greek letter sigma (Σ).
For example:
- 2 + 5 represents the addition of 2 and 5 which results in 7.
- a + b means the sum of two variables, a and b.
- For series with terms a1, a2, . . . ,an, their sum is represented as \sum_{i=0}^{n} a_i.
Sum of Numbers
When two or more numbers are added as a set of numbers that gives a total amount this process is called the addition of summation of numbers. For example, the sum of 3, 5, and 7 is 15 because 3 + 5 + 7 = 15.
In general, the sum of n numbers a1, a2, . . . ,an is calculated by adding all the individual values together,
Sum = a1 + a2 + . . . + an
Using sum operator we can add:
- One-Digit Numbers
- Two-Digit Numbers
- Square Numbers
Let's discuss the sum of each types of numbers as follows:
Sum of One-Digit Numbers
In decimal system, there are 10 digits i.e., 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. We can add these numbers very easily using counting. Let's consider an examples for sum of one-digit numbers.
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Example: Find sum of one-digit numbers are 5, 3, 6 and 7.
Solution:
Sum = 5 + 3 + 6 + 7
⇒ Sum = 21
So, the sum of the one-digit numbers 5, 3, 6, and 7 is 21.
Note: The sum of all one-digit numbers from 0 to 9 is 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45.
Sum of Two-digit Numbers
The numbers that are greater than 10 and less than 100, can be added by any numbers between them and their summation will be the total of original numbers obtained as a Sum of Two two-digit numbers.
Let's consider another example with three two-digit numbers.
Example: Find sum of two-digit numbers 15, 33 and 26.
Solution:
Sum = 15 + 33 + 26
⇒ Sum = 74
Note: The sum of all two-digit numbers from 10 to 999 is 10 + 11 + 12 + 13 + . . .+ 99 = 4905.
Sum of Square Numbers
Sum of Square Numbers refers to the sum of two numbers, each of which is raised to the power of 2 (squared). Mathematically, if we have two numbers a and b, the sum of their squares can be represented as a2 + b2.
For example, if a = 3 and b = 4, then the sum of their squares would be:
32 + 42 = 9 + 16 = 25
So, the sum of the squares of 3 and 4 is 25.
Sum of Numbers on Number Line
The total distance travelled from one point on a number line to another is represented by the sum of the numbers on the number line.
For Example: Take the numbers 2, 5 on a number line.
- Step 1: On the number line, find the first number (2).
- Step 2: Then, Shift 5 units to the right or add 5.
Thus, we only need to add these numbers together to determine their total:
= 2 + 5
= 7
Check: Speed Time Distance Formula
Sum of First n Natural Numbers
The sum of the first n natural numbers is a classic problem in mathematics, and it can be solved elegantly with a simple formula. The sum of the first n natural numbers is given by the formula:
Sn = n(n+1)/2
Where:
- Sn is the sum of the first n natural numbers.
- n is the number of terms in the sequence.
Sigma Notation for Sum
Sigma notation, represented by the Greek letter Σ (sigma), is a concise way to represent a sum of terms in mathematics. It is typically written in the form \sum_{i=0}^{n} a_i. This notation denotes the sum of the terms ai as i ranges from m to n.
Let's consider an example for sigma notation sum.
Example: Find ∑3n=1 3n2 + 9.
Solution:
∑3n=1 3n2 + 9.
= [3(1)2 + 9] + [3(2)2 + 9] + [3(3)2 + 9]
= ( 12) + (21) + ( 36)
= 69.
Therefore, the given sum evaluates to 69.
Properties of Sum
There are various properties which hold true for sum (addition), some of these properties are:
| Properties of Sum | Description |
|---|
| Closure Property | The sum of any two real numbers is always a real number. |
|---|
| Commutative Property | Changing the order of the numbers being added does not change the sum. |
|---|
| Associative Property | The way numbers are grouped in addition does not affect the sum. |
|---|
| Identity Element | The sum of any number and zero is the number itself. |
|---|
| Inverse Element | For every real number, there exists an additive inverse (negative) that, when added to the number, gives zero. |
|---|
| Distributive Property | Multiplying a number by the sum of two other numbers is the same as multiplying the number by each of the two numbers and then adding the results. |
|---|
Read More about Additive Identity Vs Multiplicative Identity.
Some of the common sum formaulas are:
Practice Problems on Sum in Maths
Problem 1: Calculate the sum of the first 10 terms of the arithmetic series where the first term, a1 is 5 and the common difference, d is 3.
Problem 2: Find the sum of the first 5 terms of a geometric series where the first term is 2 and the common ratio is 3.
Problem 3: What is the sum of all integers from 1 to 100.
Problem 4: Evaluate the sum of the series: 3+6+9+⋯+3n for n terms.
Conclusion
In conclusion, addition is the mathematical operation that helps us combine two numbers, variables, or expressions. Using addition operations, we can calculate the total bill for groceries, determine interest on a principal amount, and manage accounts. Addition is the most fundamental operation. In this article, we have explored this operation in detail, including its symbols and other methods to represent it.
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