Arithmetic progressions, commonly known as arithmetic sequences, are a fundamental concept in mathematics that appear in various fields like algebra, calculus, and computer science. They play a crucial role in understanding patterns and relationships between numbers. In this article, we will explore how to solve arithmetic progressions step by step with the assistance of maths.ai. We will begin by defining arithmetic progressions and then delve into solving them through an illustrative example.
Understanding Arithmetic Progressions
An arithmetic progression is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is known as the common difference (d). The terms of an arithmetic progression are usually denoted by ‘a’, ‘a + d’, ‘a + 2d’, and so on, where ‘a’ represents the first term.
The general formula for finding the ‘n’th term (the term at position ’n’) of an arithmetic progression is given by:
\[ a_n = a + (n-1)d \]
Where:
- \( a_n \) is the ‘n’th term
- ‘a’ is the first term
- ‘d’ is the common difference
- ’n’ is the position of the term in the sequence
Solving an Arithmetic Progression Step by Step
Let’s consider an example
Find the 10th term of an arithmetic progression where the first term is 3 and the common difference is 5.
Step 1: Identify Given Values
- First term (a) = 3
- Common difference (d) = 5
- Position of the term (n) = 10
Step 2: Apply the Formula
o calculate the ‘n’th term:
\[ a_n = a + (n-1)d \]
\[ a_{10} = 3 + (10–1) \times 5 \], \[ a_{10} = 3 + 9 \times 5 \], \[ a_{10} = 3 + 45 \]
\[ a_{10} = 48 \]
“The 10th term of the arithmetic progression is 48.”
Additional Information
the formula for the sum of the first ’n’ terms of an arithmetic progression:
\[ S_n = \frac{n}{2} \times (2a + (n-1)d) \]
Where:
- \( S_n \) is the sum of the first ’n’ terms
- ‘a’ is the first term
- ‘d’ is the common difference
- ’n’ is the number of terms
Arithmetic progressions are an essential mathematical concept, and mastering them is crucial for building a strong foundation in mathematics. With the advent of AI-powered platforms like maths.ai, students have access to immediate assistance and step-by-step guidance to solve arithmetic progression problems efficiently. The combination of well-defined formulas and the convenience of AI technology can significantly enhance students’ problem-solving skills and mathematical comprehension. Whether it’s finding specific terms or calculating sums, the power of AI in education is undeniable, offering personalized support and transforming the way students learn by making maths simple.
