# Geometric Progression - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

## Quick Facts

• Geometric Progression is considered one of the most asked concept.

• 28 Questions around this concept.

## Solve by difficulty

Fifth term of a GP is 2, then the product of its first 9 terms is:

## Concepts Covered - 4

Geometric Progression

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 $\mathrm{\mathit{a_1,a_2,a_3.....a_{n-1},a_n}}$ is in geometric progression

$\mathrm{then,\;\mathit{r=\frac{a_2}{a_1}=\frac{a_3}{a_2}=....=\frac{a_n}{a_{n-1}}}}$

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

$\\\mathrm{\mathit{a_1=a=ar^{1-1}}\;\;(1^{st}\;term)}\\\mathrm{\mathit{a_2=ar=ar^{2-1}}\;\;(2^{nd}\;term)}\\\mathrm{\mathit{a_3=ar^2=ar^{3-1}}\;\;(3^{rd}\;term)}\\...\\...\\\mathrm{\mathit{a_n=ar^{n-1}}\;\;(n^{th}\;term)}$

So, the general term or nth term of a geometric progression is $a_n=ar^{n-1}$

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.

 a a > 0 a > 0 a < 0 a < 0 r r > 1 0 < r < 1 r > 1 0 < r < 1 Result Increasing Decreasing Decreasing Increasing
Important Properties of a GP - Part 1

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 - Part 2

Important Properties of a GP

4. If $a_1,a_2,a_3.....\,\,and\,\,b_1,b_2,b_3.....$are two G.P.’s, then $a_1b_1,a_2b_2,a_3b_3.....$ and $\frac{a_1}{b_1},\frac{a_2}{b_2},\frac{a_3}{b_3}....$ are also G.P.s.

5. If $a_1,a_2,a_3,.....,a_{n-1},a_n$ are in G.P. with common ratio r, then $\log a_1,\log a_2,\log a_3...... \;\mathrm{.}$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  $\frac{a}{r^3},\frac{a}{r},ar,ar^3$.

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

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

## Study it with Videos

Geometric Progression
Important Properties of a GP - Part 1
Important Properties of a GP - Part 2

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## Books

### Reference Books

#### Geometric Progression

Mathematics for Joint Entrance Examination JEE (Advanced) : Algebra

Page No. : 5.9

Line : 40

#### Important Properties of a GP - Part 1

Mathematics Textbook for Class VII

Page No. : 5.11

Line : 1