CBSE 10
CBSE 10 Algebra: Demystify the Magical Patterns of Arithmetic Progressions
May 14, 2024
Aha
Language
English
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If you are studying algebra, one of the concepts you will undoubtedly encounter is arithmetic progression. In this guide, you will learn everything you need to know about arithmetic progressions, including notation, formulas, and types.
An arithmetic progression is a sequence of numbers in which each term is obtained by adding a fixed number to the preceding term. The fixed number is called the common difference, and it determines the rate at which the sequence increases or decreases.
For example, consider the sequence 2, 4, 6, 8, 10. This is an arithmetic progression with a common difference of 2. To get from one term to the next, you add 2.
Arithmetic progressions are typically denoted using the letter 'a'. The first term is denoted by 'a₁', the second term by 'a₂', and so on.
The first term of an arithmetic progression is denoted by 'a₁' and is the starting point of the sequence. To find the first term, you can use the formula:
a₁ = aₙ - (n - 1)d
where aₙ is the n-th term and d is the common difference.
As mentioned earlier, the common difference is the fixed number that is added to each term to get the next term in the sequence. It is denoted by 'd'. To find the common difference, you can use the formula:
d = a₂ - a₁
where a₁ is the first term and a₂ is the second term.
The general form of an arithmetic progression is:
a₁, a₁ + d, a₁ + 2d, a₁ + 3d, ...
where a₁ is the first term and d is the common difference.
There are several formulas that can be used to solve problems involving arithmetic progressions. Here are some of the most important ones:
The nth term of an arithmetic progression is given by the formula:
aₙ = a₁ + (n - 1)d
where a₁ is the first term and d is the common difference.
The sum of the first n terms of an arithmetic progression is given by the formula:
Sₙ = [2a₁ + (n - 1)d] * n / 2
where a₁ is the first term, d is the common difference, and n is the number of terms.
The sum of an arithmetic progression when the last term is given is given by the formula:
Sₙ = (a₁ + aₙ) * n / 2
where a₁ is the first term and aₙ is the last term.
There are two types of arithmetic progression: finite and infinite.
A finite arithmetic progression is a sequence of numbers that has a definite end. For example, the sequence 2, 4, 6, 8, 10 is a finite arithmetic progression because it has a last term.
An infinite arithmetic progression is a sequence of numbers that goes on forever. For example, the sequence 2, 4, 6, 8, ... is an infinite arithmetic progression because it has no last term.
Find the first term, common difference, and eighth term of the arithmetic progression 3, 6, 9, 12, ...
✅ Solution:
1. The first term a₁ is 3 and the common difference d is 3 (since 6 - 3 = 3).
2. Therefore, the eighth term is a₈ = a + (n - 1)d = 3 + (8 - 1)3 = 24.
Find the sum of the first 20 terms of the arithmetic progression 2, 5, 8, 11, ...
✅ Solution:
1. The first term a₁ is 2 andthe common difference d is 3 (since 5 - 2 = 3).
2. Therefore, the sum of the first 20 terms is Sₙ = [2a₁ + (n - 1)d] * n / 2 =
[2 * 2 + (20 - 1) * 3] * 20 / 2 = 610.
The general form of an arithmetic progression is
a₁, a₁ + d, a₁ + 2d, a₁ + 3d, ...
where a₁ is the first term and d is the common difference.
An arithmetic progression is a sequence of numbers in which each term is obtained by adding a fixed number to the preceding term. For example, 2, 4, 6, 8, ... is an arithmetic progression with a common difference of 2.
To find the sum of an arithmetic progression, you can use the formula:
Sₙ = [2a₁ + (n - 1)d] * n / 2
where a₁ is the first term, d is the common difference, and n is the number of terms.
There are three types of progressions in mathematics: arithmetic progression, geometric progression, and harmonic progression.
Arithmetic progressions are an important concept in algebra and mathematics in general. By understanding the notation, formulas, and types of arithmetic progressions, you can solve a wide variety of problems. If you want to learn more about maths, check out Aha AI for more resources and courses.
