## 05 - Three-digit Numbers not Divisible by 3

**Problem**

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

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**Problem**

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

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**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?

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.

**Arithmetic Progression, AP**

Definition

Examples of arithmetic progression are:

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

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**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