An arithmetic progression is a sequence of numbers where the difference between any two consecutive terms is constant. This constant difference is called the common difference, and it allows the terms of the sequence to be generated in a predictable pattern.
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The 'n'th term of an arithmetic progression can be calculated using the formula: 'a_n = a_1 + (n-1)d', where 'a_1' is the first term and 'd' is the common difference.
The sum of the first 'n' terms of an arithmetic progression can be calculated using the formula: 'S_n = \frac{n}{2}[2a_1 + (n-1)d]', where 'a_1' is the first term and 'd' is the common difference.
Arithmetic progressions can be used to model a variety of real-world situations, such as the growth of a savings account, the depreciation of an asset, or the distance traveled in a constant-speed motion.
The difference between any two terms in an arithmetic progression is a multiple of the common difference, and the ratio of any two terms is constant.
Arithmetic progressions are a fundamental concept in mathematics and have applications in fields such as finance, physics, and computer science.
Review Questions
Explain the relationship between the common difference and the terms of an arithmetic progression.
The common difference is the constant value that is added to each term in an arithmetic progression to generate the next term. This means that the difference between any two consecutive terms in the progression is always equal to the common difference. For example, if the first term is 'a_1' and the common difference is 'd', then the second term is 'a_2 = a_1 + d', the third term is 'a_3 = a_2 + d = a_1 + 2d', and so on. The common difference allows the terms of the sequence to be generated in a predictable pattern.
Describe how the explicit formula for an arithmetic progression can be used to calculate the 'n'th term of the sequence.
The explicit formula for an arithmetic progression is 'a_n = a_1 + (n-1)d', where 'a_1' is the first term and 'd' is the common difference. This formula allows you to calculate the 'n'th term of the sequence directly, without having to generate all the intermediate terms. For example, if the first term is 'a_1 = 3' and the common difference is 'd = 2', then the 10th term can be calculated as 'a_{10} = 3 + (10-1)2 = 3 + 18 = 21'. The explicit formula is a powerful tool for working with arithmetic progressions, as it enables you to quickly determine any term in the sequence without having to go through the entire list.
Explain how the sum formula for an arithmetic progression can be used to find the total of the first 'n' terms of the sequence.
The sum formula for an arithmetic progression is 'S_n = \frac{n}{2}[2a_1 + (n-1)d]', where 'a_1' is the first term, 'd' is the common difference, and 'n' is the number of terms. This formula allows you to calculate the sum of the first 'n' terms of the sequence without having to add up all the individual terms. For example, if the first term is 'a_1 = 5', the common difference is 'd = 3', and you want to find the sum of the first 20 terms, you can use the formula to calculate 'S_{20} = \frac{20}{2}[2(5) + (20-1)(3)] = 10[10 + 57] = 10(67) = 670'. The sum formula is particularly useful when working with large arithmetic progressions, as it provides a efficient way to determine the total of the sequence.
The explicit formula for an arithmetic progression is a formula that allows you to calculate the 'n'th term of the sequence directly, given the first term and the common difference.