Problems on Geometric Series

Question 31

Question:  

Answer:   

Question 32

Question:  

Answer:   

Question 33

Question:   i. Show that the arithmetic mean of two positive numbers is greater than or equal to their geometric mean.
ii. The arithmetic mean of two positive integers a and b (a>b)
is twice their Geometric mean. Prove that

iii. Construct a quadratic equation in x such that AM of its roots is A and GM is G.

Answer:    i. Let a and b be the numbers.












iii. Let a and b be the numbers. A be the AM and G be the GM


The quadratic equation in x is

Question 34

Question:   i. If a, b, c are in GP, prove that log a, log b, log c are in AP.

AP.


in AP.


vi. If a, b, c are in GP and x, y be the AM's of a, b and b, c respectively, prove that

Answer:   
Taking log on both sides





















Let A = first term, R = Common ratio







(dividing both sides by xy)

Question 35

Question:  

Answer:    Let g₁, g₂, g₃,----g_n be the GM's between a and b.

Question 36

Question:   The vibration of a group are damped so that the amplitude of successive deflection are in GP. If the amplitude is first 12cm and then 8cm. Find the amplitude of 6^th Vibration.

Answer:    Here the successive amplitudes are in GP
ar = 8

Question 37

Question:   If S₁,S₂,S₃------S_n are the sums of n terms of n GP's, whose common ratio are respectively 1,2,3,--------- Show that,

Answer:   

Question 38

Question:  

Answer:   

Question 39

Question:   If S₁, S₂, S₃,....., S_n are the sums of infinite GP's whose first

Answer:   

This is an AP with a = 2, d = 1 and n = n

Question 40

Question:  
Find the first term.

Answer:   


\ The first term is 16.