Geometric Progression

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

A.   413,633 C.   489,335
B.   498,533 D.   431,336

 

Problem
In a fund raising show, a group of philanthropists agreed that the first one to arrive would pay 25¢ to enter, and each later would pay twice as much as the preceding person. The total amount collected from all of them was \$262,143.75. How many of them paid?

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

 

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.
 

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.
 

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?
 

Geometric progression with some given terms

Situation
The 4th term of a geometric progression is 6 and the 10th 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
 

Arithmetic, geometric, and harmonic progressions

Elements
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 of the 1st n terms
 

Derivation of Formula for the Future Amount of Ordinary Annuity

The sum of ordinary annuity is given by
 

$F = \dfrac{A[ \, (1 + i)^n - 1 \, ]}{i}$

 

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

Derivation

Figure for Derivation of Sum of Ordinary Annuity

 

$F = \text{ Sum}$

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

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

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.
 

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