02 - Numbers 4, 2, 5, and 18 are Added Respectively to the First Four Terms of AP, Forming Into a GP Jhun Vert Fri, 04/17/2020 - 04:09 pm

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?
 

01 - Geometric Progression With Some Given Terms Jhun Vert Fri, 04/17/2020 - 04:07 pm

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
 

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.
 

Derivation of Sum of Finite and Infinite Geometric Progression

Geometric Progression, GP
Geometric progression (also known as geometric sequence) is a sequence of numbers where the ratio of any two adjacent terms is constant. The constant ratio is called the common ratio, r of geometric progression. Each term therefore in geometric progression is found by multiplying the previous one by r.
 

Eaxamples of GP:

  • 3, 6, 12, 24, … is a geometric progression with r = 2
  • 10, -5, 2.5, -1.25, … is a geometric progression with r = -1/2