Arithmetic Progression Concepts

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.