Problems on Geometric Progression- Test Questions

Question 41

Question:   If the p^th, q^th, r^th, s^th terms of an AP are in GP, show that
p - q, q - r, r - s are in GP.

Answer:    Let a = first term and d = common difference of the AP.




Since a_p, a_q, a_r, a_s are in GP, then

Question 42

Question:   If p, q, r are in AP and x, y, z are in GP, then prove that

Answer:   
Also q = p + d, r = p + 2d, where d is the common difference of the AP.
x, y, z are in GP. Let R be the common ratio

Question 43

Question:   Find geometrically the GM between a and b?

Answer:    Let G be the Geometric mean between a and b.

Geometrically, if AB = a, BC = b.
On a line ABC, draw a semi circle on AC, with AC as diameter. Draw BD perpendicular to AC at B to cut the semicircle at D.
Draw AD and CD.




BD is the GM between AB and BC.

Question 44

Question:   Insert 3 GM's between 8 and 648.

Answer:    a = 8, b = 648 R = Common ratio
Let the three GM be aR, aR², aR³. then b = aR⁴

\ The three GM's are 8(3), 8(3)², 8(3)³

Question 45

Question:   If one arithmetic mean A and two geometric means g₁, g₂ be

Answer:    Let the two numbers be a and b.
If A is the AM between a and b then a, A, b are in AP.

If g₁, g₂ are the GM's between a and b then a, g₁, g_2, b are in GP.

Question 46

Question:   If S₁, S₂, S₃ be respectively the sum of n, 2n, 3n terms of GP, prove that:
i) S₁²+S₂² = S₁(S₂ + S₃)
ii) S₁(S₃ - S₂) = (S₂ - S₁)²

Answer:   

Question 47

Question:   Find the sum of n terms of the series 5 + 55+ 555 + ...

Answer:   


Here a = 10, r = 10, n = n

Question 48

Question:   If y = x + x² + x³+ ... to infinite number of terms, then

Answer:   

Question 49

Question:   Insert
i. 2 Geometric means between 2 and 16

iii. 4 Geometric means between 5 and 0.00005
iv. 5 Geometric means between 3 and 192

Answer:   

Question 50

Question:   If the AM between two numbers exceeds their GM by 2 and the ratio of two numbers is 4, find the numbers.

Answer:    Let the numbers be a and b (a > b)

A - G = 2 (by data)