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This section introduces arithmetic progressions (APs) as sequences of numbers where each term is derived by adding a constant difference to the preceding term. It covers key concepts such as the structure of APs, how to determine the nth term, and formulas for calculating the sum of the first n terms of an AP.
In this section, an arithmetic progression (AP) is defined as a sequence where each term after the first is obtained by adding a constant, called the common difference, to the previous term. The general form of an AP is expressed as a, a + d, a + 2d, ..., where 'a' is the first term and 'd' is the common difference. To identify whether a sequence is an AP, one must check that the differences between successive terms remain constant. The nth term of an AP can be calculated using the formula a_n = a + (n - 1)d, helping us find any term in the sequence. Additionally, the sum of the first n terms is calculated using the formula S_n = (2a + (n - 1)d) / 2. When the last term (l) is known, the sum can also be calculated as S_n = n(a + l) / 2. This section lays the groundwork for understanding the significance of APs in various practical applications.
Arithmetic Progression: A sequence where each term after the first is obtained by adding a constant.
Common Difference: The fixed difference between consecutive terms in an AP.
nth Term Formula: Used to find any term in the AP.
Sum of Terms: Calculating the total of the first n terms using specific formulas.
In a line, the terms align, add 'd' to find, all is fine!
Imagine a rabbit hopping along a path, discovering new spots by adding a constant distance 'd' each time. This hopping path forms an arithmetic progression.
AP = Add Pattern; always add a constant difference 'd'.
Example 1: In an AP where a = 3 and d = 2, the first five terms are 3, 5, 7, 9, 11.
Example 2: To find the 10th term of an AP with first term 2 and common difference 4, use a_10 = 2 + (10 - 1) * 4 = 38.
Term: Arithmetic Progression (AP)
Definition: A sequence of numbers where each term after the first is obtained by adding a fixed number, called the common difference.
A sequence of numbers where each term after the first is obtained by adding a fixed number, called the common difference.
Term: Common Difference
Definition: The fixed number added to each term to achieve the next term in an AP.
The fixed number added to each term to achieve the next term in an AP.
Term: nth Term
Definition: The term at position 'n' in a sequence, calculated using a specific formula in the case of APs.
The term at position 'n' in a sequence, calculated using a specific formula in the case of APs.
Term: Sum of Terms
Definition: The total of a specified number of terms in an AP, calculated using specific sum formulas.
The total of a specified number of terms in an AP, calculated using specific sum formulas.