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Questions and Answers
What is the formula for finding the arithmetic mean of an arithmetic progression (AP)?
What is the formula for finding the arithmetic mean of an arithmetic progression (AP)?
In an arithmetic progression, if the first term is 3 and the common difference is 4, what is the third term?
In an arithmetic progression, if the first term is 3 and the common difference is 4, what is the third term?
What does the formula $a_n = a + (n-1)d$ represent in an arithmetic progression?
What does the formula $a_n = a + (n-1)d$ represent in an arithmetic progression?
Which field uses arithmetic progressions to model compound interest?
Which field uses arithmetic progressions to model compound interest?
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How is the sum of the first n terms of an arithmetic progression calculated?
How is the sum of the first n terms of an arithmetic progression calculated?
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What is the formula to find the n-th term of an Arithmetic Progression (AP)?
What is the formula to find the n-th term of an Arithmetic Progression (AP)?
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For the AP: 3, 8, 13, 18, ..., what is the sum of the first 5 terms?
For the AP: 3, 8, 13, 18, ..., what is the sum of the first 5 terms?
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In the AP: 2, 5, 8, 11, ..., what is the common difference?
In the AP: 2, 5, 8, 11, ..., what is the common difference?
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What is the 15th term of the AP: 4, 10, 16, 22, ...?
What is the 15th term of the AP: 4, 10, 16, 22, ...?
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For an AP where $a = 2$, the common difference is -3, what is the sum of the first 8 terms?
For an AP where $a = 2$, the common difference is -3, what is the sum of the first 8 terms?
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Study Notes
Arithmetic Progression
An Arithmetic Progression (AP) is a sequence of numbers where the common difference between any two consecutive terms is constant. The general form of an AP is given by:
(a_1, a_1 + d, a_1 + 2d, a_1 + 3d, \ldots, a_1 + (n-1)d)
where (a_1) is the first term and (d) is the common difference.
N-th Term of an AP
The formula for finding the n-th term of an AP is:
(a_n = a + (n-1)d)
where (a) is the first term, (d) is the common difference, and (n) is the number of terms.
Example
Find the n-th term of the AP: 1, 2, 3, 4, 5, 6, 7, 8, 9, (\ldots).
Solution: Given, (a = 1) and (d = 2-1 = 1). By the formula, (a_n = 1 + (n-1)1). For example, (a_{10} = 1 + (10-1)1 = 10).
Common Difference
The common difference in an AP is the constant difference between consecutive terms. It can be positive, negative, or zero.
Example
Find the common difference of the AP: 1, 4, 7, 10, 13, (\ldots).
Solution: Given, (a = 1) and (d = 4-1 = 3). By the formula, (d = a_2 - a_1 = a_3 - a_2 = \ldots = a_n - a_{n-1}).
Sum of N Terms of an AP
The sum of the first n terms of an AP is given by:
(S_n = \frac{n}{2}[2a + (n-1)d])
where (a) is the first term, (d) is the common difference, and (n) is the number of terms.
Example
Find the sum of the first 10 terms of the AP: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, (\ldots).
Solution: Given, (a = 1) and (d = 2-1 = 1). By the formula, (S_{10} = \frac{10}{2}[2(1) + (10-1)(1)] = 5(2+9) = 50).
Arithmetic Mean
The arithmetic mean of an AP is the average of the first and last terms. It is given by:
(AM = \frac{a_1 + a_n}{2})
where (a_1) is the first term and (a_n) is the last term.
Example
Find the arithmetic mean of the AP: 2, 5, 8, 11, 14, (\ldots).
Solution: Given, (a_1 = 2) and (a_n = 14). By the formula, (AM = \frac{2 + 14}{2} = 8).
Applications of AP
APs have various applications in different fields, including finance, physics, and engineering. They are used to model compound interest, calculate the average speed of an object, and determine the sum of the terms in a series.
Example
A person invests $1000 in a bank that offers an annual interest rate of 5%. If the interest is compounded annually, what is the value of the investment after 10 years?
Solution: Given, (a = 1000), (d = 5%), and (n = 10). By the formula, (a_{10} = a + (n-1)d = 1000 + (10-1)(5%) = 1100). Therefore, the value of the investment after 10 years is $1100.
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Description
Test your understanding of Arithmetic Progression (AP) concepts such as n-th term, common difference, sum of terms, arithmetic mean, and applications. Practice calculating AP elements and solving related problems.
