Geometric Progression(GP)-Nth term Formula, First Term & Common Ratio, Solved Examples, FAQs

🔢 What is a Geometric Progression (GP)?

A Geometric Progression (GP) is a sequence of numbers where each term is obtained by multiplying the previous term by a constant factor called the common ratio (r).

📌 General Form of a GP : a, ar, ar², ar³,…

Where:

• a = First term
• r = Common ratio (ratio between consecutive terms)
• n = Number of terms

📌 Formula for the nth Term of a GP:

a_n = a⋅r⁽ⁿ⁻¹⁾

This formula helps find any term in the sequence without listing all the terms.

For example, 2, 4, 8, 16, 32, 64, … is a GP, where the common ratio is 2.

Similarly,

Consider a series 1, 1/2, 1/4, 1/8, 1/16, …

In the given examples, the ratio is a constant. Such sequences are called Geometric Progressions. It is abbreviated as G.P.

In general

\(\begin{array}{l}\text{A sequence}\ a_{1},a_{2},a_{3},…,a_{n},… \text{is a G.P, then}\ \frac{a_{k+1}}{a_{k}} = r \ \ \ [k > 1]\end{array} \)

Where r is a constant which is known as common ratio and none of the terms in the sequence is zero.

Proof : The n^th term of the Geometric series is denoted by a_n and the elements of the sequence are written as a₁, a₂, a_3, a₄, …, a_n.

a₁ = a_,
a₂ = a*r
a₃ = a*r²
a₄ = a*r³
a_n = a*r^n-1

Therefore, the formula to find the nth term of GP is:

an = tn = arn-1

Note: The nth term is the last term of finite GP.

Common Ratio (r) of GP

Consider the sequence a, ar, ar², ar³,……

First term = a

Second term = ar

Third term = ar²

Similarly, nth term, t_n = ar^n-1

Thus, the common ratio of geometric progression formula is given as:

Common ratio = (Any term) / (Preceding term)

= t_n / t_n-1

= (ar^{n – 1} ) /(ar^{n – 2})

= r

Thus, the general term of a GP is given by ar^n-1 and the general form of a GP is a, ar, ar²,…..

For Example: r = t₂ / t₁ = ar / a = r

Condition for the given sequence to be a geometric sequence:

For any sequence to be considered a GP the ratio of two any successive terms must remain constant:

a₂/a₁ = a₃/a₂ =  … = a_n/a_n-1 = r (common ratio).

General term or nth term of a Geometric Sequence a, ar, ar², ar³, ar⁴ is given by : 

a_n = ar^n-1

where, 

a₁ = first term, 
a₂ = second term
a_n = last term (or the nth term)

Nth Term from the Last Term is given by:

a_n = l/r^n-1

where, l is the last term.

Types of GP

GP is further classified into two types, which are:

1. Finite Geometric Progression (Finite GP)
2. Infinite Geometric Progression (Infinite GP)

Finite Geometric Progression

Finite G.P. is a sequence that contains finite terms in a sequence and can be written as a, ar, ar², ar³,……ar^n-1, arⁿ. 

An example of Finite GP is 1, 2, 4, 8, 16,……512

Infinite Geometric Progression

Infinite G.P. is a sequence that contains infinite terms in a sequence and can be written as a, ar, ar², ar³,……ar^n-1, arⁿ……, i.e. it is a sequence that never ends.

Examples of Infinite GP are:

• 1, 2, 4, 8, 16,……..
• 1, 1/2, 1/4, 1/8, 1/16,………

Properties of Geometric Progression (GP)

Some of the important properties of GP are listed below:

• Three non-zero terms a, b, c are in GP if and only if b² = ac
• If y² = xz, then the three non-zero terms x, y, and z are in GP.
• In a GP,
  Three consecutive terms can be taken as a/r, a, ar
  Four consecutive terms can be taken as a/r³, a/r, ar, ar³
  Five consecutive terms can be taken as a/r², a/r, a, ar, ar²
• Square of a term in GP is product of its adjacent terms. a²_k = a_k-1 × a_k+1
• In a finite GP, the product of the terms equidistant from the beginning and the end is the same
  That means, t₁.t_n = t₂.t_n-1 = t₃.t_n-2 = …..
• If we multiply or divide a non-zero quantity by each term of the GP, then the resulting sequence is also in GP with the same common difference. Please note addition and subtraction does not keep a GP as GP. It is true for AP only.
• The product and quotient of two GP’s is again a GP
• If each term of a GP is raised to the power by the same non-zero quantity, the resultant sequence is also a GP.
• Geometric progressions typically means either exponential growth or exponential decay, depending on whether the common ratio 0 < r < 1 or r > 1. This makes it useful in modeling things like population growth, radioactive decay, and interest compounding.
• If we select terms at regular intervals (say every kth term) from a GP, those terms also form an GP with a common ratio d^k. For example, if we select every third terms from 1, 2, 4, 8, 16, 32, 64, … we get 1, 8, 64, … which is again an GP with common ration as 8.
• If a₁, a₂, a₃,… is a GP of positive terms then log a₁, log a₂, log a₃,… is an AP (arithmetic progression) and vice versa. For example, if the terms of the GP are a,ar,ar²,ar³,… the logarithms of these terms will be log⁡a, log⁡a+log⁡r, log⁡a+2log⁡r, … which forms an AP.

Difference between Arithmetic Sequence and Geometric Sequence

	Arithmetic Sequence	Geometric Sequence
Definition	A sequence in which the difference between any two consecutive terms is constant.	A sequence in which the ratio of any two consecutive terms is constant.
Common Term	The common difference is denoted as ‘d’.	The common ratio is denoted as ‘r’.
General Formula	The nth term is given by an​=a1+(n−1)d, where a1​ is the first term and ‘d’ is the common difference.	The nth term is given by, an​=a1×r(n−1), where a1​ is the first term and ‘r’ is the common ratio.
Example	2, 5, 8, 11, 14, … (Here, d = 3)	3, 6, 12, 24, 48, … (Here, r = 2)
Nature of Growth	Linear growth: The terms increase or decrease by a constant amount.	Exponential growth: The terms increase or decrease by a constant factor.
Graph Appearance	Forms a straight line when plotted on a graph.	Forms a curve (exponential growth or decay) when plotted on a graph.
Sum of n Terms	Given by Sn​= n​/2[2a1 +(n−1)d]	Given by Sn = a1(rn -1)/(r – 1)

Solved Examples on Geometric Progression (GP)

Example 1: Suppose the first term of a GP is 4 and the common ratio is 5, then the first five terms of GP are?

First term, a = 4
Common ratio, r = 5
Now, the first five term of GP is
a, ar, ar², ar³, ar⁴
a = 4
ar = 4 × 5 = 20
ar² = 4 × 25 = 100
ar³ = 4 × 125 = 500
ar⁴ = 4 × 625 = 2500
Thus, the first five terms of GP with first term 4 and common ratio 5 are:
4, 20, 100, 500, and 2500

Example 2: If 3, 9, 27,…., is the GP, then find its 9th term.

nth term of GP is given by:

a_n = ar^n-1

given, GP 3, 9, 27,….
Here, a = 3 and r = 9/3 = 3
Therefore,
a₉ = 3 x 3^{9 – 1}
    = 3 × 6561
    = 19683

Example 3: If the first term is 10 and the common ratio of a GP is 3, then write the first five terms of GP.

Solution: Given,

First term, a = 10

Common ratio, r = 3

We know the general form of GP for first five terms is given by:

a, ar, ar², ar³, ar⁴

a = 10

ar = 10 × 3 = 30

ar² = 10 × 3² = 10 × 9 = 90

ar³ = 10 × 3³ = 270

ar⁴ = 10 × 3⁴ = 810

Therefore, the first five terms of GP with 10 as the first term and 3 as the common ratio are:

10, 30, 90, 270 and 810

Example 4: If 2, 4, 8,…., is the GP, then find its 10th term.

Solution: The nth term of GP is given by:

2, 4, 8,….

Here, a = 2 and r = 4/2 = 2

a_n = ar^n-1

Therefore,

a₁₀ = 2 x 2^{10 – 1}

= 2 × 2⁹

= 1024

Practice Problems on Geometric Progression

1. Find the equivalent fraction of the recurring decimal 0.595959…..
2. What is the 12th term of the sequence 4, -8, 16, -64,….?
3. Check whether the given sequence is GP.
   27, 9, 3, …
4. Write the first five terms of a GP whose first term is 3 and the common ratio is 2.

📌 Applications of GP in JEE & Board Exams:

✅ Compound Interest Calculations
✅ Population Growth & Decay Problems
✅ Physics (Radioactive Decay, Waves)

Frequently Asked Questions on Geometric Progression

Q1

What is a Geometric Progression?

Geometric Progression (GP) is a type of sequence where each succeeding term is produced by multiplying each preceding term by a fixed number, which is called a common ratio.

Q2

Give an example of Geometric Progression.

The example of GP is: 3, 6, 12, 24, 48, 96,…

Q3

What is the general form of GP?

The general form of a Geometric Progression (GP) is given by a, ar, ar², ar³, ar⁴,…,ar^n-1

a = First term

r = common ratio

ar^n-1 = nth term

Q4

What is the common ratio in GP?

The common multiple between each successive term and preceding term in a GP is the common ratio. It is a constant value that is multiplied by each term to get the next term in the Geometric series. If a is the first term and ar is the next term, then the common ratio is equal to:

ar/a = r

Q5

What is not a geometric progression?

If the common ratio between each term of a geometric progression is not equal then it is not a GP.

Q6

Can the values of ‘a’ and ‘r’ be 0?

No, the value of a≠0, if the first term becomes zero, the series will not continue. Similarly, r≠0.

Mastering Geometric Progression is essential for Board Exams & JEE Mains/Advanced! 🚀