Geometric and Harmonic Progression - Comparison between AP GP HP, Proof of Sum of General and Infinite GP, Increasing and Decreasing GP Subtypes

In the previous video I have completed the basics of Arithmetic Progression and in this video I have completed explaining the basics of Geometric and Harmonic Progression. Definition of Geometric Progression - The number in a series are said to be in Geometric Progression when the numbers increase or decrease by a constant by multiplying them with the particular constant. Then I have explained if I have to assume 3 or 4 or 5 numbers that are in Geometric Progression for solving a question then how we will assume them. After that I have explained how a geometric mean is inserted between two numbers. Then I have explained Increasing and Decreasing Geometric Progressions and their subtypes. After that I have proved the formula of sum of a general Geometric Progression having n terms. Then told what is the sum of an infinite Geometric Progression. Then I have taken up Geometric Progression question solving. After that I have started with Harmonic Progressions and proved a property which says that if a,b,c are in Harmonic Progression then a/c = (a-b)/(b-c). Then I have shown how to insert a Harmonic Mean between two numbers a and b which is equal to 2ab/(a+b). After that I have compared the values of Arithmetic Mean, Geometric Mean and Harmonic Mean of two numbers a and b which are positive real numbers and a is not equal to b. Questions discussed by me in this video are listed below. Your comments, doubts and suggestions are welcome in the comment section below.

Question 1. If the 5th term of a Geometric Progression is 81 and the first term is 16, what will be the fourth term of the same Geometric Progression ?

Question 2. The 4th and 10th terms of a Geometric Progression are 1/3 and 243 respectively. Find the second term ?

Question 3. Two numbers X and Y are such that their Geometric Mean is 20 % less than their Arithmetic Mean. Find the ratio of X to Y ?

Question 4. In an infinite Geometric Progression, each term is equal to three times the sum of the terms that follow. If the first term of the series is 8, find the sum of the series ?

Question 5. If m,n and p are in Geometric Progression, then log m, log n and log p are in ??

Question 6. The sum of an infinite Geometric Progression is 3 and the sum of an infinite Geometric Progression formed from the square of the terms of the original series is 6. Find the first term of the first series ?

Question 7. After striking the floor, an elastic ball rebounds to 4/5th of the height from which it has fallen. Find the total distance that it travels before coming to rest if it has been gently dropped from a height of 120 meters ?

Question 8. The sum of an infinite Geometric Progression whose common ratio is numerically less than 1 is 32 and the sum of the first two terms is 24. What will be the third term ?

Question 9. If the ratio of Harmonic Mean of two numbers to their Geometric Mean is 12:13, find the ratio of the numbers ?

Question 10. There is a Harmonic Progression whose 7th term is 1/10 and 12th term is 1/25 . Find the 20th term of the Harmonic Series ?

Question 11. If x,y,z are in Harmonic Progression, then show that x/y+z , y/x+z and z/x+y are also in Harmonic Progression.

Question 12. If x²,y² and z² are in Arithmetic Progression, then prove that y+z, z+x, x+y are in Harmonic Progression.