Geometric Progression - Practice Questions & MCQ

A geometric sequence is a sequence where the first term is non-zero and the ratio between consecutive terms is always constant. The ‘constant factor’ is called the common ratio and denoted by ‘r’ . r is also a non-zero number.

The first term of a G.P. is usually denoted by 'a'.

If  is in geometric progression

Eg,

• 2, 6, 18, 54, .... (a = 2, r = 3)
• 4, 2, 1, 1/2, 1/4, .... ( a = 4, r = 1/2)
• -5, 5, -5, 5, .......( a = -5, r = -1)

General Term of a GP

If ‘a’ is the first term and ‘r’ is the common ratio, then 

So, the general term or n^th term of a geometric progression is

Increasing and Decreasing GP

For a GP to be increasing or decreasing, r > 0. If r < 0, then the terms of G.P. are alternately positive and negative so neither increasing nor decreasing.

Important Properties of a GP

1. If a , b, c are in GP, then b² = a.c

2. If each term of a G.P. is multiplied by a fixed constant or divided by a non-zero fixed constant then the resulting series is also in G.P. with same common ratio as the orginal series.

3. If each term of a G.P. is raised to some real number m, then the resulting series is also in G.P.

Important Properties of a GP

    4. If are two G.P.’s, then  and  are also G.P.s.

    5. If  are in G.P. with common ratio r, then is in A.P. and converse also holds true.

    6. If three numbers in G.P. whose product is given are to be taken, then take them as a/r, a, ar. 

    7. If four numbers in G.P. whose product is given are to be taken, then take them as  .

    8. Product of terms equidistant from start and end of the G.P. is constant and it equals product of first and the last terms.

Some questions based on Geometric Progression

Q: If the product of three terms of a GP is 512, and sum of their product in pair is 224, then the sum of three numbers will be

Sol: It is given that three terms are in G.P. and their product is 512,

    So take three numbers as a/r, a, ar

    Then, the product, a/r・a・ar = 512  ⇒ a³ = 512 ⇒ a = 8

    Now,

    Sum of the product in the pair is 224

    So, a/r・a + a・ar + a/r・ar = 224

    ⇒  a² (1/r + r + 1) = 224      

    Put the value of a,

    ⇒ 64 (1/r + r + 1) = 224

    ⇒ (1/r + r + 1) = 7/2

    ⇒ r = 2 or r = ½

    So, the three numbers which are in GP is 4, 8, 16 or 16, 8, 4

    Sum is 16 + 8 + 4 = 28