# Arithmetic Progression

## Numbers 4, 2, 5, and 18 are Added Respectively to the First Four Terms of AP, Forming Into a GP

**Problem**

If 4, 2, 5, and 18 are added respectively to the first four terms of an arithmetic progression, the resulting series is a geometric progression. What is the common difference of the arithmetic progression?

## General Term of Arithmetic Sequence that Models the Potential Annual Salaries

**Problem**

A job posted at jobstreet.com 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*= 38,400 + 1600

_{n}*n*

B.

*a*= 33,400 + 2600

_{n}*n*

C.

*a*= 36,400 + 1400

_{n}*n*

D.

*a*= 34,400 +1800

_{n}*n*

## Three-digit numbers not divisible by 3

**Problem**

How many three-digit numbers are not divisible by 3?

## Arithmetic, geometric, and harmonic progressions

**Elements***a*_{1} = value of the first term*a*_{m} = value of any term after the first term but before the last term*a _{n}* = value of the last term

*n*= total number of terms

*m*=

*m*

^{th}term after the first but before

*n*

^{th}

*d*= common difference of arithmetic progression

*r*= common ratio of geometric progression

*S*= sum of the 1

^{st}

*n*terms

## 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

Below is the derivation of this relationship.

## Derivation of Sum of Arithmetic Progression

**Arithmetic Progression, AP**

Definition

*d*.

Examples of arithmetic progression are:

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