# Geometric Progression

**Problem**

The first three terms of a geometric progression are 2*x*, 4*x* + 14 and 20*x* - 14. Find the sum of the first ten terms.

A. 413,633 | C. 489,335 |

B. 498,533 | D. 431,336 |

- Read more about Sum of the first ten terms of a Geometric Progression
- Log in or register to post comments
- 270 reads

**Problem**

Smith and Jones, both 50% marksmen, decide to fight a duel in which they exchange alternate shots until one is hit. What are the odds in favor of the man who shoots first?

A. 1/3 | C. 2/3 |

B. 1/2 | D. 1/4 |

- Read more about Duel of Two 50% Marksmen: Odds in favor of the man who shoots first
- Log in or register to post comments
- 2772 reads

## Sum of Areas of Infinite Number of Squares

**Problem**

The side of a square is 10 m. A second square is formed by joining, in the proper order, the midpoints of the sides of the first square. A third square is formed by joining the midpoints of the second square, and so on. Find the sum of the areas of all the squares if the process will continue indefinitely.

- Read more about Sum of Areas of Infinite Number of Squares
- Log in or register to post comments
- 8733 reads

## Sum of Areas of Equilateral Triangles Inscribed in Circles

**Problem**

An equilateral triangle is inscribed within a circle whose diameter is 12 cm. In this triangle a circle is inscribed; and in this circle, another equilateral triangle is inscribed; and so on indefinitely. Find the sum of the areas of all the triangles.

- Read more about Sum of Areas of Equilateral Triangles Inscribed in Circles
- Log in or register to post comments
- 6853 reads

## 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?

- Read more about Numbers 4, 2, 5, and 18 are Added Respectively to the First Four Terms of AP, Forming Into a GP
- Log in or register to post comments
- 7337 reads

## Geometric progression with some given terms

**Situation**

The 4^{th} term of a geometric progression is 6 and the 10^{th} term is 384.

Part 1: What is the common ratio of the G.P.?

A. 1.5

B. 3

C. 2.5

D. 2

Part 2: What is the first term?

A. 0.75

B. 1.5

C. 3

D. 0.5

Part 3: What is the seventh term?

A. 24

B. 32

C. 48

D. 96

- Read more about Geometric progression with some given terms
- Log in or register to post comments
- 6700 reads

## 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

- Read more about Arithmetic, geometric, and harmonic progressions
- Log in or register to post comments
- 83733 reads

## Derivation of Formula for the Future Amount of Ordinary Annuity

The sum of ordinary annuity is given by

To learn more about annuity, see this page: ordinary annuity, deferred annuity, annuity due, and perpetuity.

### Derivation

$F = \text{ Sum}$

$F = A + F_1 + F_2 + F_3 + \cdots + F_{n-1} + F_n$

$F = A + A(1 + i) + A(1 + i)^2 + A(1 + i)^3 + \cdots + A(1 + i)^{n-1} + A(1 + i)^n$

- Read more about Derivation of Formula for the Future Amount of Ordinary Annuity
- Log in or register to post comments
- 13468 reads

## 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.

- Read more about Relationship Between Arithmetic Mean, Harmonic Mean, and Geometric Mean of Two Numbers
- Log in or register to post comments
- 104591 reads