Number System || Sum of Series ? (LESSON-5)

Summary

In this lesson, the concept of the sum of series is explored, with a focus on natural numbers, odd numbers, even numbers, and their squares and cubes. The video explains five key formulas used to calculate the sum of series, highlighting formulas for natural numbers, squares, cubes, odd, and even numbers. The presenter emphasizes the simplicity of the formulas and provides step-by-step examples to demonstrate their application. The lesson also includes strategies for handling series that start from numbers other than 1, and offers practical advice for competitive exam preparation.

Takeaways

• 😀 The sum of series is an important concept that can be categorized into two main models: natural numbers and odd/even/different numbers.
• 😀 To solve sum of series problems, it is essential to memorize and apply five formulas: three for natural numbers and two for odd/even numbers.
• 😀 The formula for the sum of natural numbers from 1 to n is: n * (n + 1) / 2.
• 😀 For the sum of squares of natural numbers from 1 to n, the formula is: n * (n + 1) * (2n + 1) / 6.
• 😀 To find the sum of cubes of natural numbers from 1 to n, use the formula: (n * (n + 1) / 2) ^ 2.
• 😀 The sum of odd numbers can be found using the formula: (n + 1 / 2) ^ 2, where n is the last odd number in the series.
• 😀 The sum of even numbers can be calculated using the formula: (n / 2) * (n / 2 + 1).
• 😀 When the series does not start from 1, such as starting from 51 or 10, use subtraction to find the sum (total of 1 to n minus the sum of the previous set).
• 😀 For non-standard series like squares starting from 10 to 20, the formula for sum of squares can be applied by subtracting the sum of squares from 1 to 9 from the sum of squares from 1 to 20.
• 😀 It is important to identify the correct model (natural numbers, odd/even numbers, etc.) and apply the corresponding formula to solve the sum of series problems in competitive exams.

Q & A

What is the sum of series, and why is it important?

The sum of series refers to adding a sequence of numbers to find the total value. It's important because it helps in simplifying calculations for long series, especially in exams, where manual addition would be too time-consuming.

How are the sum of series questions categorized?

The sum of series questions are categorized into two models: Model 1 focuses on natural numbers (1, 2, 3, ...), while Model 2 deals with odd, even, and other specific number types.

What is the formula for the sum of natural numbers?

For the sum of natural numbers, the formula is: n * (n + 1) / 2, where 'n' is the last number in the series.

What is the difference between the sum of squares and the sum of cubes in a series?

The sum of squares of natural numbers uses the formula: n * (n + 1) * (2n + 1) / 6, while the sum of cubes uses the formula: (n * (n + 1) / 2)².

How do you calculate the sum of odd numbers in a series?

For odd numbers, the sum is calculated using the formula: x², where x = (n + 1) / 2. The value of 'n' is the last number in the series.

What is the formula for the sum of even numbers in a series?

The sum of even numbers is calculated using the formula: x * (x + 1), where x = n / 2, and 'n' is the last even number in the series.

What formula is used for series starting from non-1 numbers, like 10² to 20²?

For series starting from non-1 numbers, like 10² to 20², you calculate the sum from 1 to the last number (20²), and then subtract the sum from 1 to the previous number (9² in this case).

How do you approach a sum of series question for odd numbers, such as 51, 53, 55, ..., 99?

For odd numbers, first find the total sum from 1 to the last odd number (99), then subtract the sum from 1 to the previous odd number (49).

Why is it not necessary to memorize every possible power formula like one raised to the fourth power, two to the fourth power, etc.?

You don't need to memorize higher powers like one to the fourth power or two to the fourth power unless you're preparing for exams like CAT. For most exams, understanding the basic formulas is sufficient.

What should you remember when solving sum of series questions in exams?

When solving sum of series questions, always identify the model of the series (natural, odd, even, or others), select the appropriate formula, and then apply it correctly to find the sum.