Geometric Progression

Derivation of Product of First n Terms of Geometric Progression

The product of the first $n$ terms of a Geometric Progression is given by the following:

Given the first term $a_1$ and last term $a_n$:

$P_n = \sqrt{(a_1 \times a_n)^n}$


Given the first term $a_1$ and the common ratio $r$

$P_n = {a_1}^n \times r^{n(n - 1)/2}$


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


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?

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

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

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

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

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

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.


Figure for Derivation of Sum of Ordinary Annuity


$F = \text{ Sum}$

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

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


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