Geometric Progression (GP)

Geometric Progression (GP)

Edited By Komal Miglani | Updated on Jul 02, 2025 05:54 PM IST

A geometric progression, also known as a geometric sequence, is a mathematical sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with a common ratio of 3. In real life, we use geometric progressions to calculate the size of exponential population growth, such as bacteria in a container.

This Story also Contains
  1. What is Geometric Progression?
  2. Increasing and Decreasing GP
  3. Properties of Geometric Progression
  4. Solved Examples Based on Geometric Progression
Geometric Progression (GP)
Geometric Progression (GP)

In this article, we will cover the concept of the Geometric Progression. This category falls under the broader category of Sequence and series, which is a crucial Chapter in class 11 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination(JEE Main) and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more. Over the last ten years of the JEE Main exam (from 2013 to 2023), a total of 16 questions have been asked on this concept, including one in 2013, one in 2017, two in 2019, three in 2020, four in 2021, two in 2022 and four in 2023

What is Geometric Progression?

A geometric progression or 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 is denoted by ‘r’. r is also a non-zero number.

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

If $a_1, a_2, a_3 \ldots . a_{n-1}, a_n$ is in geometric progression then, $r=\frac{a_2}{a_1}=\frac{a_3}{a_2}=\ldots . .=\frac{a_n}{a_{n-1}}$

Eg,
- $2,6,18,54, \ldots .(a=2, r=3)$
- $4,2,1,1 / 2,1 / 4, \ldots .(a=4, r=1 / 2)$
- $-5,5,-5,5, \ldots \ldots .(a=-5, r=-1)$

General Term of a GP

If ' $a$ ' is the first term and ' $r$ ' is the common ratio, then
$
\begin{aligned}
& a_1=a=a r^{1-1}\left(1^{\text {st }} \text { term }\right) \\
& a_2=a r=a r^{2-1}\left(2^{\text {nd }} \text { term }\right) \\
& a_3=a r^2=a r^{3-1}\left(3^{\text {rd }} \text { term }\right) \\
& \cdots \\
& \cdots \\
& a_n=a r^{n-1}\left(\mathrm{n}^{\text {th }} \text { term }\right)
\end{aligned}
$

So, the general term or $\mathrm{n}^{\text {th }}$ term of a geometric progression is $a_n=a r^{n-1}$

Increasing and Decreasing GP

For a GP to be increasing or decreasing, r > 0. Since, If r < 0, then the terms of G.P. are alternately If a > 0, then G.P. is increasing if r > 1 and decreasing if 0 <r<1.

If a < 0 then G.P. is decreasing If r > and increasing if 0 < r <1positive 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

Properties of Geometric Progression

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 the same common ratio as the original 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.

4. If two series are in GP then the the product of the series is also in GP.

If $a_1, a_2, a_3 \ldots$ and $b_1, b_2, b_3 \ldots$...are two G.P.'s, then $a_1 b_1, a_2 b_2, a_3 b_3 \ldots \ldots$ and $\frac{a_1}{b_1}, \frac{a_2}{b_2}, \frac{a_3}{b_3} \ldots$ are also G.P.
5. If $a_1, a_2, a_3, \ldots ., a_{n-1}, a_n$ are in G.P. with common ratio $r$, then is in A.P. and the converse also holds true.
6. If three numbers in G.P. whose product is given are to be taken, then take them as $\mathrm{a} / \mathrm{r}, \mathrm{a}$, ar.
7. If four numbers in G.P. whose product is given are to be taken, then take them as $\frac{a}{r^3}, \frac{a}{r}, a r, a r^3$.

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

Recommended Video Based on Geometric Progression


Solved Examples Based on Geometric Progression

Example 1: Let $0<z<y<x$ be three real numbers such that $\frac{1}{x}, \frac{1}{y}, \frac{1}{z}$ are in an arithmetic progression and $x, \sqrt{2} y, z$ are in a geometric progression. If $x y+y z+z x=\frac{3}{\sqrt{2}} x y z$, then $3(x+y+z)^2$ [JEEMAINS 2023]
Solution:
$
\begin{aligned}
& 2 y^2=x z \\
& \frac{2}{y}=\frac{x+z}{x z}=\frac{x+z}{2 y^2} \quad 4 y^2+2 y^2=\frac{3}{\sqrt{2}} y \cdot 2 y^2 \\
& x+z=4 y \quad 6 y^2=3 \sqrt{2} y^3 \\
& x y+y z+z x=\frac{3}{\sqrt{2}} x y z \quad \begin{array}{l}
y=\sqrt{2} \\
x+y+z=5 y=5 \sqrt{2}
\end{array} \\
& y(x+z)+z x=\frac{3}{\sqrt{2}} x z \cdot y^{3(x+y+z)^2}=3 \times 50=150 \\
&
\end{aligned}
$

Hence, the required answer is 150.

Example 2: For the two positive numbers $\mathrm{a}, \mathrm{b}$ is $\mathrm{a}$, and $\mathrm{b}$ and $\frac{1}{18}$ are in a geometric progression, while $\frac{1}{\mathrm{a}}, 10$ and $\frac{1}{\mathrm{~b}}$ are in an arithmetic progression, then $16 \mathrm{a}+\mathrm{b}$ is equal to [JEE MAINS 2023]
Solution:
$
\begin{aligned}
& \mathrm{b}^2=\frac{a}{18} \\
& 20=\frac{1}{a}+\frac{1}{b} \\
& a=\frac{b}{20 b-1} \\
& b^2=\frac{1}{18} \times \frac{b}{20 b-1} \\
& 360 b^2-18 b-1=0 \\
& 360 b^2-30 b+12 b-1=0 \\
& (12 b-1)(30 \mathrm{~b}+1)=0 \\
& b=\frac{1}{12}, \frac{-1}{30}(\text { rejected }) \\
& a=\frac{1}{8} \\
& 16 \mathrm{a}+12 \mathrm{~b}=2+1=3
\end{aligned}
$

Hence, the required answer is 3.

Example 3: Let the first term a and the common ratio r of a geometric progression be positive integers. If the sum of squares of its first three is 33033, then the sum of these terms is equal to : [JEE MAINS 2023]

Solution:

Let $a, a r, a r^2$ be three terms of GP
Given : $\mathrm{a}^2+(a r)^2+\left(\mathrm{ar}^2\right)^2=33033$
$
\begin{aligned}
& a^2\left(1+r^2+r^4\right)=11^2 \cdot 3.7 .13 \\
& \Rightarrow \mathrm{a}=11 \text { and } 1+r^2+r^4=3.7 .13 \\
& \Rightarrow r^2\left(1+r^2\right)=273-1 \\
& \Rightarrow r^2\left(r^2+1\right)=272=16 \times 17 \\
& \Rightarrow r^2=16 \\
& \therefore r=4 \quad[\because r>0]
\end{aligned}
$

The sum of three terms $=a+a r+a r^2+a=\left(1+r+r^2\right)$
$
\begin{aligned}
& =11(1+4+16) \\
& =11 \times 21=231
\end{aligned}
$

Hence, the required answer is 231.

Example 4: If $\mathrm{a}_1(>0), \mathrm{a}_2, \mathrm{a}_3, \mathrm{a}_4, \mathrm{a}_5$ are in a G.P., $\mathrm{a}_2+\mathrm{a}_4=2 \mathrm{a}_3+1$ and $3 \mathrm{a}_2+\mathrm{a}_3=2 \mathrm{a}_{4,}$, then $\mathrm{a}_2+\mathrm{a}_4+2 \mathrm{a}_5$ is equal to [JEE MAINS 2022]
Solution:
$
\begin{aligned}
& \mathrm{a}_1, \mathrm{a}_2, \ldots, \mathrm{a}_5 \longrightarrow \mathrm{G} \cdot \mathrm{p} \\
& \mathrm{a}_2+\mathrm{a}_4=2 \mathrm{a}_3+1 ; \quad 3 \mathrm{a}_2+\mathrm{a}_3=2 \mathrm{a}_4 \\
& \mathrm{ar}+\mathrm{ar}^3=2 \mathrm{a}^2+1 ; 3 \mathrm{ar}+\mathrm{ar}^2=2 \mathrm{ar}^3 \\
& 2 \mathrm{r}^2-\mathrm{r}-3=0 \\
& \mathrm{r}=-1 \cdot \frac{3}{2}
\end{aligned}
$

Now
$
\begin{aligned}
& \frac{3 \mathrm{a}}{2}+\frac{27 \mathrm{a}}{8}=\frac{3 \mathrm{a}}{2}+1 \\
& \therefore \quad \mathrm{n}=\frac{3}{2} \\
& \mathrm{a}=\frac{8}{3} \\
& \mathrm{a}_2+\mathrm{a}_4+2 \mathrm{a}_5=\mathrm{ar}+\mathrm{ax}^3+2 \mathrm{ax}^4 \\
& =40
\end{aligned}
$

Hence, the required answer is 40.

Example 5: Let $\mathrm{A}_1, \mathrm{~A}_2, \mathrm{~A}_3, \cdots$ be an increasing geometric progression of positive real numbers. If If $\mathrm{A}_1 \mathrm{~A}_3 \mathrm{~A}_5 \mathrm{~A}_7=\frac{1}{1296}$ and $\mathrm{A}_2+\mathrm{A}_4=\frac{7}{36}$, then the value of $\mathrm{A}_6+\mathrm{A}_8+\mathrm{A}_{10}$ is equal to [JEE MAINS 2022]

Solution:
$
\begin{aligned}
& \text { Let } \mathrm{A}_1=\mathrm{a} \& \text { common ratio=r. } \\
& \mathrm{a} \cdot \mathrm{ar}^2 \cdot \mathrm{ar}^4 \cdot \mathrm{ar}^6=\frac{1}{1296} \Rightarrow \mathrm{a}^4 \mathrm{r}^{12}=\frac{1}{1296} \Rightarrow \mathrm{a}^2 \mathrm{r}^6=\frac{1}{36} \Rightarrow \mathrm{ar}^2=\frac{1}{6} \\
& \Rightarrow \mathrm{A}_4=\frac{1}{6}, \quad \mathrm{~A}_2+\mathrm{A}_4=\frac{7}{36} \\
& \Rightarrow \mathrm{A}_2=\frac{7}{36}-\frac{1}{6}=\frac{1}{36}=\mathrm{ar} \Rightarrow \mathrm{r}^2=6 \Rightarrow \mathrm{r}=\sqrt{6}, \mathrm{a}=\frac{1}{36 \sqrt{6}} \\
& A_6+A_8+A_{10}=\operatorname{ar}^5+\operatorname{ar}^7+\operatorname{ar}^9=\frac{1}{36 \sqrt{6}}\left(6^2 \sqrt{6}+6^3 \sqrt{6}+6^4 \sqrt{6}\right) \\
& =1+6+6^2=43 \\
&
\end{aligned}
$

Hence, the required answer is 43.

Frequently Asked Questions (FAQs)

1. What happens to a GP if the common ratio is between 0 and 1?
If the common ratio is between 0 and 1 (0 < r < 1), the terms of the GP will decrease in absolute value, approaching zero as the sequence progresses.
2. What happens to a GP if the common ratio is greater than 1?
If the common ratio is greater than 1 (r > 1), the terms of the GP will increase indefinitely as the sequence progresses.
3. What is the effect of changing the common ratio in a GP?
Changing the common ratio in a GP alters the rate at which the terms grow or decay. A larger ratio (in absolute value) leads to faster growth or decay, while a smaller ratio leads to slower growth or decay.
4. What is the relationship between exponential functions and GPs?
Exponential functions and GPs are closely related. The terms of a GP can be expressed as an exponential function of their position: an = a1 * r^(n-1), which is an exponential function of n.
5. Can a GP with negative terms converge?
Yes, a GP with negative terms can converge. The convergence depends on the absolute value of the common ratio being less than 1, regardless of whether the terms are positive or negative.
6. What is the common ratio in a GP?
The common ratio in a GP is the constant factor by which each term is multiplied to obtain the next term. It can be found by dividing any term by the previous term in the sequence.
7. Can a GP have a common ratio of 1?
Yes, a GP can have a common ratio of 1. In this case, all terms in the sequence will be equal to the first term, resulting in a constant sequence.
8. What is the sum formula for a finite GP?
The sum of n terms of a GP is given by the formula: Sn = a1(1-r^n)/(1-r) for r ≠ 1, and Sn = na1 for r = 1, where Sn is the sum, a1 is the first term, r is the common ratio, and n is the number of terms.
9. What does it mean for a GP to converge?
A GP converges when the sum of its terms approaches a finite limit as the number of terms increases indefinitely. This occurs when the absolute value of the common ratio is less than 1 (|r| < 1).
10. Can a GP have negative terms?
Yes, a GP can have negative terms. This can occur if the first term is negative, or if the common ratio is negative, or both.
11. How can you convert a repeating decimal to a fraction using GP concepts?
Repeating decimals can be expressed as the sum of an infinite GP. For example, 0.333... can be written as 3/10 + 3/100 + 3/1000 + ..., which is a GP with a1 = 3/10 and r = 1/10. Using the infinite sum formula, this equals 1/3.
12. What is a Geometric Progression (GP)?
A Geometric Progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In other words, the ratio between any two consecutive terms is constant throughout the sequence.
13. How is a GP different from an Arithmetic Progression (AP)?
In a Geometric Progression, each term is multiplied by a constant ratio to get the next term, while in an Arithmetic Progression, each term is added to a constant difference to get the next term. GPs involve multiplication, while APs involve addition.
14. What is a geometric mean in relation to a GP?
The geometric mean of a set of numbers is the nth root of their product, where n is the number of terms. In a GP, any term (except the first and last) is the geometric mean of the terms immediately before and after it.
15. How do you find the nth term of a GP?
The nth term of a GP can be found using the formula: an = a1 * r^(n-1), where an is the nth term, a1 is the first term, r is the common ratio, and n is the position of the term.
16. How do you find the common ratio if you're given two terms of a GP?
To find the common ratio when given two terms of a GP, divide the later term by the earlier term. For example, if am and an are two terms where m < n, then r = (an/am)^(1/(n-m)).
17. What is the sum formula for an infinite GP?
The sum of an infinite GP is given by the formula: S∞ = a1/(1-r), where |r| < 1. This formula only applies when the absolute value of the common ratio is less than 1, ensuring that the series converges.
18. How can you insert geometric means between two given terms?
To insert n geometric means between two terms a and b, first calculate the common ratio r = (b/a)^(1/(n+1)). Then, the geometric means are ar, ar^2, ar^3, ..., ar^n.
19. How do you find the sum of a GP when given the first term, last term, and number of terms?
When given the first term (a1), last term (an), and number of terms (n), you can use the formula: Sn = (n/2)(a1 + an). This is derived from the standard sum formula by substituting for the common ratio.
20. How do you find the term number of a specific value in a GP?
To find the term number n for a specific value an in a GP, use the formula: n = log(an/a1) / log(r) + 1, where a1 is the first term and r is the common ratio. This is derived from solving the nth term formula for n.
21. How can you use GPs to approximate square roots?
GPs can be used in iterative methods to approximate square roots. For example, the Babylonian method for finding √a starts with an initial guess x0 and generates a sequence where xn+1 = (xn + a/xn)/2, which converges to √a.
22. What is the significance of the golden ratio in GPs?
The golden ratio (φ ≈ 1.618) appears in a special GP where the ratio of successive terms approaches φ. This GP is related to the Fibonacci sequence and has applications in art, architecture, and nature.
23. How can GPs be used in computer algorithms, particularly in divide-and-conquer strategies?
Many divide-and-conquer algorithms, like quicksort or merge sort, split problems into smaller subproblems. The sizes of these subproblems often form a GP, which is crucial in analyzing the time complexity of these algorithms.
24. What is the role of GPs in financial mathematics, particularly in annuities?
In financial mathematics, GPs are used to model annuities where payments grow at a constant rate. The present value of such an annuity is calculated using the sum formula of a GP, taking into account the growth rate and discount rate.
25. What is the role of GPs in the theory of continued fractions?
In the theory of continued fractions, the convergents (approximations to the fraction) often have numerators and denominators that form recursive sequences. In some cases, these sequences are related to GPs, especially for quadratic irrationals.
26. How do you determine if a sequence is a GP?
To determine if a sequence is a GP, check if the ratio between any two consecutive terms is constant. If this ratio (common ratio) is the same for all pairs of consecutive terms, the sequence is a GP.
27. Can a GP have zero as a term?
A GP can have zero as a term, but only if it's the first term (a1 = 0) or if a term becomes zero due to rounding in practical applications. If any non-first term is exactly zero, all subsequent terms will also be zero.
28. What is the product formula for n terms of a GP?
The product of n terms of a GP is given by the formula: P = (a1)^n * r^(n(n-1)/2), where P is the product, a1 is the first term, r is the common ratio, and n is the number of terms.
29. How does changing the first term affect a GP?
Changing the first term of a GP scales all terms in the sequence by the same factor. If you multiply the first term by k, all terms in the GP will be multiplied by k, but the common ratio remains unchanged.
30. Can a GP have complex numbers as terms?
Yes, a GP can have complex numbers as terms. The common ratio in this case would also be a complex number, and the usual formulas for GPs still apply.
31. How is a GP related to exponential growth or decay?
A GP directly models exponential growth or decay. If the common ratio r > 1, it represents exponential growth; if 0 < r < 1, it represents exponential decay. The GP formula an = a1 * r^(n-1) is equivalent to the exponential function form.
32. What is the significance of the geometric series 1 + 1/2 + 1/4 + 1/8 + ...?
This geometric series with a1 = 1 and r = 1/2 is significant because it converges to 2. It's often used to introduce the concept of infinite series convergence and has applications in various fields, including fractals and computer science.
33. What is the connection between GPs and logarithms?
Logarithms and GPs are closely related. Taking logarithms of terms in a GP results in an arithmetic progression. This relationship is the basis for slide rules and is useful in solving certain types of exponential equations.
34. What is a harmonic progression and how is it related to a GP?
A harmonic progression is a sequence where the reciprocals of the terms form an arithmetic progression. It's related to a GP because if you take the reciprocals of terms in a GP, you get a harmonic progression.
35. How can you use GPs to model population growth?
GPs can model population growth when the population increases by a constant percentage in each time period. If P0 is the initial population and r is the growth rate, the population after n periods is given by Pn = P0(1+r)^n, which follows a GP.
36. What is the role of GPs in compound interest calculations?
GPs are fundamental in compound interest calculations. The amount A after n periods with principal P, interest rate r per period, is given by A = P(1+r)^n. This forms a GP with first term P and common ratio (1+r).
37. What is a geometric sequence of vectors?
A geometric sequence of vectors is a sequence where each vector is obtained by multiplying the previous vector by a scalar (the common ratio). The magnitude of the vectors forms a standard GP, while their directions remain constant or rotate uniformly.
38. What is the connection between GPs and fractals?
Many fractals, such as the Cantor set or Sierpinski triangle, can be described using GPs. The scaling factor between successive iterations in these fractals often follows a GP, contributing to their self-similar structure.
39. How do GPs appear in music theory?
In music theory, the frequencies of notes in an equal-tempered scale form a GP. Each semitone increase multiplies the frequency by the common ratio 2^(1/12), allowing for 12 semitones to double the frequency (one octave).
40. What is the role of GPs in probability theory, particularly in geometric distributions?
The geometric distribution in probability theory is based on a GP. It models the number of trials needed to get the first success in a series of independent Bernoulli trials, where the probability of each outcome forms a GP.
41. How can you use GPs to model radioactive decay?
Radioactive decay can be modeled using a GP with a common ratio less than 1. If N0 is the initial amount of a radioactive substance and r is the fraction remaining after each time period, the amount after n periods is N = N0 * r^n.
42. How do GPs relate to the concept of half-life in physics and chemistry?
Half-life, the time taken for a quantity to reduce to half its initial value, is directly related to GPs. In a decay process modeled by a GP, the common ratio r after one half-life period is 1/2, and the sequence of measurements at successive half-lives forms a GP with r = 1/2.
43. What is the connection between GPs and Pascal's triangle?
In Pascal's triangle, the entries along any diagonal form a GP. For example, the sequence 1, 3, 6, 10, 15, ... along a diagonal is not a GP, but its differences 2, 3, 4, 5, ... form an AP, and the reciprocals of these differences 1/2, 1/3, 1/4, 1/5, ... form a GP.
44. How do GPs appear in optics, particularly in the design of multi-layer optical coatings?
In optics, multi-layer coatings often use layers with thicknesses forming a GP. This design helps in creating broadband reflectors or anti-reflection coatings, where the ratio of successive layer thicknesses is constant.
45. What is the significance of GPs in number theory, particularly in the study of Mersenne primes?
Mersenne primes are prime numbers of the form 2^n - 1. The sequence of numbers 2^n forms a GP with first term 1 and common ratio 2. The study of when 2^n - 1 is prime involves properties of this GP.
46. How can GPs be used to model the spread of information or rumors in social networks?
The spread of information in social networks can sometimes be modeled using GPs, especially in the early stages. If each person shares information with a constant number of new people in each time step, the total number of informed individuals can form a GP.
47. What is the connection between GPs and the binary number system?
The place values in the binary number system form a GP with first term 1 and common ratio 2. Each bit represents a power of 2 (1, 2, 4, 8, 16, ...), which is a GP. This connection is fundamental to understanding binary representation in computing.
48. How do GPs appear in the study of fractional dimensions, such as in the Hausdorff dimension of fractals?
In fractal geometry, the Hausdorff dimension often involves analyzing how the number of self-similar pieces scales as the size is reduced. This scaling often follows a GP, where the ratio of the number of pieces to the scaling factor determines the fractal dimension.
49. How can GPs be used to model the resolution of digital imaging systems?
In digital imaging, the resolution of sensors or displays often follows a GP. For example, common resolutions like 640x480, 1280x960, 2560x1920 form a GP with a common ratio of 2 in each dimension. This relates to the concept of pixel density and image scaling.
50. What is the significance of GPs in the study of chaos theory and dynamical systems?
In chaos theory, the period-doubling route to chaos involves a sequence of bifurcations where the period of oscillation doubles. The points at which these doublings occur often form a GP, leading to the Feigenbaum constant, a universal scaling factor in many chaotic systems.

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