arithmetic progression

General Term of Arithmetic Sequence that Models the Potential Annual Salaries

A job posted at offered a starting salary of \$40,000 per year and guaranteeing a raise of \$1600 per year for the rest of 5 years. Write the general term for the arithmetic sequence that models potential annual salaries.

A.   an = 38,400 + 1600n
B.   an = 33,400 + 2600n
C.   an = 36,400 + 1400n
D.   an = 34,400 +1800n

Arithmetic, geometric, and harmonic progressions

a1 = value of the first term
am = value of any term after the first term but before the last term
an = value of the last term
n = total number of terms
m = mth term after the first but before nth
d = common difference of arithmetic progression
r = common ratio of geometric progression
S = sum

Arithmetic Progression, AP

Arithmetic progression is a sequence of numbers in which the difference of any two adjacent terms is constant. The constant difference is commonly known as common difference and is denoted by d. Examples of arithmetic progression are as follows:

Relationship Between Arithmetic Mean, Harmonic Mean, and Geometric Mean of Two Numbers

For two numbers x and y, let x, a, y be a sequence of three numbers. If x, a, y is an arithmetic progression then 'a' is called arithmetic mean. If x, a, y is a geometric progression then 'a' is called geometric mean. If x, a, y form a harmonic progression then 'a' is called harmonic mean.

Let AM = arithmetic mean, GM = geometric mean, and HM = harmonic mean. The relationship between the three is given by the formula

$AM \times HM = GM^2$


Below is the derivation of this relationship.

Derivation of Sum of Arithmetic Progression

Arithmetic Progression, AP

Arithmetic Progression (also called arithmetic sequence), is a sequence of numbers such that the difference between any two consecutive terms is constant. Each term therefore in an arithmetic progression will increase or decrease at a constant value called the common difference, d.

Examples of arithmetic progression are:

  • 2, 5, 8, 11,... common difference = 3
  • 23, 19, 15, 11,... common difference = -4


Derivation of Formulas