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Sum – Definition with Examples

Sum is a crucial cornerstone in mathematics and plays a significant role in the world of numbers. As a building block for numerous mathematical operations, the sum represents the combined total of two or more numbers. Here at Brighterly, we are dedicated to helping young learners comprehend the ins and outs of mathematical concepts. In this knowledge, we will delve into various sum types such as the sum of two-digit numbers, the sum of first n natural numbers, and more. Additionally, we will examine how to find sums on a number line, provide you with solved examples, and offer practice problems to boost your understanding. So, let’s embark on this exciting journey into the world of sums!

What is Sum?

A sum is the result of adding two or more numbers together. It is an essential concept in arithmetic and plays a vital role in various branches of mathematics. The process of finding the sum is called addition, and the numbers being added are called addends. The symbol “+” is used to represent the addition operation. For example, the sum of 4 and 5 is 9, which is represented as 4 + 5 = 9.

Sum of Two-digit Numbers

Adding two-digit numbers involves combining the numbers in the tens and ones place separately. Follow these steps to find the sum of two-digit numbers:

1. Add the digits in the ones place.
2. If the sum is greater than 9, carry the digit in the tens place to the next column.
3. Add the digits in the tens place, along with any carried digit from the previous step.
4. Write the sum obtained in the tens place.

For example, let’s find the sum of 27 and 46:

Sum of Three-Digit Numbers

The process of adding three-digit numbers is similar to that of two-digit numbers, but with an additional place value – the hundreds place. Follow these steps:

1. Add the digits in the ones place.
2. If the sum is greater than 9, carry the digit in the tens place to the next column.
3. Add the digits in the tens place, along with any carried digit from the previous step.
4. If the sum is greater than 9, carry the digit in the hundreds place to the next column.
5. Add the digits in the hundreds place, along with any carried digit from the previous step.
6. Write the sum obtained in the hundreds place.

For example, let’s find the sum of 253 and 178:

Sum of First n Natural Numbers

The sum of the first n natural numbers can be found using a simple formula:

Sum = (n * (n + 1)) / 2

Here, “n” represents the last natural number in the series. This formula is derived from the arithmetic progression concept, with the first term being 1, the last term being n, and the common difference between consecutive terms being 1.

For example, to find the sum of the first 10 natural numbers:

Sum = (10 * (10 + 1)) / 2 = (10 * 11) / 2 = 55

Sum of Odd numbers formula

The sum of the first n odd numbers can be calculated using the following formula:

Sum = n²

Here, “n” represents the number of odd numbers in the series. This formula is derived from the arithmetic progression concept, with the first term being 1, the last term being (2n – 1), and the common difference between consecutive terms being 2.

For example, to find the sum of the first 5 odd numbers:

Sum = 5² = 25

Sum of Even numbers formula

The sum of the first n even numbers can be calculated using the following formula:

Sum = n * (n + 1)

Here, “n” represents the number of even numbers in the series. This formula is derived from the arithmetic progression concept, with the first term being 2, the last term being 2n, and the common difference between consecutive terms being 2.

For example, to find the sum of the first 5 even numbers:

Sum = 5 * (5 + 1) = 5 * 6 = 30

Sum of Squares of n Natural Numbers

The sum of the squares of the first n natural numbers can be found using this formula:

Sum = (n * (n + 1) * (2n + 1)) / 6

Here, “n” represents the last natural number in the series.

For example, to find the sum of the squares of the first 5 natural numbers:

Sum = (5 * (5 + 1) * (2 * 5 + 1)) / 6 = (5 * 6 * 11) / 6 = 55

Sum of Cubes of n Natural Numbers

The sum of the cubes of the first n natural numbers can be calculated using the following formula:

Sum = (n² * (n + 1)²) / 4

Here, “n” represents the last natural number in the series.

For example, to find the sum of the cubes of the first 4 natural numbers:

Sum = (4² * (4 + 1)²) / 4 = (16 * 25) / 4 = 100

Finding the Sum On Number Line

A number line is a visual representation of numbers on a straight line, with equally spaced intervals. It can be used to find the sum of two or more numbers by starting at the first number and moving along the number line according to the value of the second number.

For example, to find the sum of 3 and 5 using a number line:

1. Start at the number 3 on the number line.
2. Move 5 spaces to the right (in the positive direction).
3. The number you land on is the sum, which is 8 in this case.

Solved Examples

1. Find the sum of 48 and 79:

2. Find the sum of the first 7 natural numbers:

Sum = (7 * (7 + 1)) / 2 = (7 * 8) / 2 = 28

3. Find the sum of the first 6 odd numbers:

Sum = 6² = 36

Practice Problems

1. Find the sum of 154 and 378.
2. Find the sum of the first 20 natural numbers.
3. Find the sum of the first 8 even numbers.
4. Find the sum of the squares of the first 10 natural numbers.

Conclusion

Frequently Asked Questions on Sum

What is the sum of the first 100 natural numbers?

To find the sum of the first 100 natural numbers, we can use the formula for the sum of first n natural numbers:

Sum = (n * (n + 1)) / 2

In this case, n = 100, so the formula becomes:

Sum = (100 * (100 + 1)) / 2 = (100 * 101) / 2 = 5050

Therefore, the sum of the first 100 natural numbers is 5050. This method, known as the arithmetic progression formula, allows us to quickly find the sum without having to manually add each number.

How do you find the sum of the first n even numbers?

To find the sum of the first n even numbers, we use the following formula:

Sum = n * (n + 1)

Here, “n” represents the number of even numbers in the series. This formula is derived from the arithmetic progression concept, with the first term being 2, the last term being 2n, and the common difference between consecutive terms being 2.

For example, to find the sum of the first 5 even numbers:

Sum = 5 * (5 + 1) = 5 * 6 = 30

Thus, the sum of the first 5 even numbers is 30.

What is the sum of the first 15 odd numbers?

To find the sum of the first n odd numbers, we use the following formula:

Sum = n²

Here, “n” represents the number of odd numbers in the series. This formula is derived from the arithmetic progression concept, with the first term being 1, the last term being (2n – 1), and the common difference between consecutive terms being 2.

For example, to find the sum of the first 15 odd numbers:

Sum = 15² = 225

Therefore, the sum of the first 15 odd numbers is 225. Using this formula, we can quickly calculate the sum without having to add each odd number individually.

Information Sources

• Wikipedia: Summation
• Wolfram MathWorld: Sum
• National Council of Educational Research and Training (NCERT): Mathematics Textbooks