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The section explains what Arithmetic Progressions are, including their definition, common difference, and how to derive their nth term and sum of the first n terms. It also provides practical examples and applications in real-life scenarios.
Arithmetic Progressions (AP) are sequences of numbers where the difference between consecutive terms is constant, referred to as the common difference (d). The section emphasizes that the first term is denoted as 'a' and the nth term can be calculated using the formula:
$$a_n = a + (n-1)d$$
Examples such as salary increment patterns and physical measurements illustrate AP in real life. The chapter further explores identifying whether a sequence is an AP by checking if the differences between consecutive terms are equal. The section concludes with how to find the sum of the first n terms in an AP using the formula:
$$S_n = \frac{n}{2} [2a + (n-1)d]$$
This foundational knowledge is critical for solving various mathematical and real-world problems involving sequences.
Arithmetic Progression: A sequence formed by repeated addition of a fixed number.
Common Difference: The consistent difference between consecutive terms.
Formula for nth term: a_n = a + (n-1)d
Sum of n terms: S_n = n/2 [2a + (n-1)d]
In an AP, terms don't stray, they grow by d each day.
Imagine climbing a staircase with equally spaced steps, each step higher represents the addition of the common difference.
Remember: AAP - Always Add the Progression (A for 'Arithmetic', A for 'Add', P for 'Progression').
In a salary increment scenario, if a person earns $1000 initially and receives a $100 increment yearly, the sequence of salaries forms an AP: 1000, 1100, 1200, ...
A ladder where the distance between rungs decreases consistently can be seen as an AP.
Term: Arithmetic Progression (AP)
Definition: A sequence of numbers in which the difference between consecutive terms is constant.
A sequence of numbers in which the difference between consecutive terms is constant.
Term: Common Difference (d)
Definition: The fixed amount added to each term to get the next term in an AP.
The fixed amount added to each term to get the next term in an AP.
Term: First Term (a)
Definition: The initial term in an arithmetic progression.
The initial term in an arithmetic progression.
Term: nth Term
Definition: The term which is in the position n in a sequence.
The term which is in the position n in a sequence.
Term: Sum of First n Terms (S_n)
Definition: The sum of the first n terms in an arithmetic progression.
The sum of the first n terms in an arithmetic progression.