A Geometric Progression (GP) is a sequence of numbers where each term is found by multiplying the previous term by a fixed number called the common ratio. This sequence follows a clear pattern, making it easy to understand and apply in different situations.
For example, in sequence 2, 4, 8, 16, 32, each number is obtained by multiplying the previous number by 2. Here, the common ratio is 2.
Geometric progressions appear in many real-life situations. The branches of a tree can follow a geometric pattern if each branch splits into two new ones.
In finance, an investment that grows by a fixed percentage every year forms a geometric sequence. Even pyramid stacking in games or competitions follows this pattern when each level has a set number of times more objects than the previous one.
This fun math concept appears in many real-life places—from tree branches to video game scores!
Let’s learn about geometric progression in a simple and exciting way.
What is a Geometric Sequence?
A geometric sequence is a series of numbers where each term is obtained by multiplying the previous term by a constant, called the common ratio ( r ). For example, if the first term ( a ) is 2, and the common ratio ( r ) is 3, the sequence will be: 2, 6, 18, 54, 162, .. Here’s how it works:
- Start with the first term, a = 2.
- Multiply the first term by the common ratio to get the next term: 2 × 3 = 6.
- Multiply the second term by the common ratio to get the third term: 6 × 3=18.
- Keep repeating this process: 18 × 3 = 54, 54 × 3 = 162, and so on.
To find the common ratio, divide any term by the term before it: 6/2 = 3, 18/6 = 3, 54/3 = 18, and so on. This confirms the common ratio is constant.
What is a Geometric Progression?
Geometric progression meaning is simple: it’s a sequence of numbers where we multiply each number by the same value to get the next number. This value is called the common ratio.
Example: In the sequence 2, 4, 8, 16, 32...
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Multiply 2 by 2 = 4
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4 × 2 = 8
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8 × 2 = 16
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16 × 2 = 32
So, the common ratio is 2, and this is a geometric progression.
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General Form of Geometric Progression (GP)
The general form of a geometric progression is: a, ar,ar 2 ,ar 3 ,… For instance, if a = 5. and r = 2, the sequence becomes: 5, 10, 20, 40, 80,... Each term is obtained by repeatedly multiplying by r.
Properties of Geometric Progression
Here are the key properties of a geometric progression:
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Middle Number Rule: In 2, 4, 8 → 4² = 2 × 8
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Same Product Rule: 2 × 16 = 4 × 8
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Scaling Numbers: 3, 6, 12 becomes 6, 12, 24 if you multiply by 2
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Flip Rule: Taking reciprocals of 2, 4, 8 gives 1/2, 1/4, 1/8 (still a GP)
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Power Rule: Squaring 2, 4, 8 gives 4, 16, 64 (still a GP)
These help you spot and understand patterns better.
Growth or Decay : If the numbers in a GP keep getting bigger, it means the common ratio is greater than 1. If the numbers get smaller, the common ratio is between 0 and 1. For e xample: 2,4,8 (growing) and 8,4,2 (shrinking). Skipping Numbers : If you pick every few numbers (like every 2nd or 3rd number) from a GP, the result is still a GP. For e xample: From 2, 4, 8, 16, 32, picking every 2nd number gives 2, 8, 32, which is another GP.
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Types of Geometric Progression
Geometric Progression can be categorized into two main types:
- Finite Geometric Progression (Finite GP)
- Infinite Geometric Progression (Infinite GP)
Let’s explore the types of Geometric Progression (GP) with examples below:
Finite Geometric Progression (Finite GP)
A finite GP has a fixed number of terms in the sequence. It starts with the first term a and ends at the n -th term, which is a × r n−1
Example:
Consider a=1 and r=2, with 10 terms: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512 Here, the sequence stops after 10 terms, making it a finite GP .
Infinite Geometric Progression (Infinite GP)
An infinite GP has no end; the sequence continues indefinitely. It starts with the first term aa a and keeps multiplying by the common ratio rr r . Examples: For a=1 , r=2: 1, 2, 4, 8, 16, 32,… This sequence grows infinitely as r>1 For a=1, r=1/2 1,1/2,1/4,1/8,1/16,… This sequence becomes smaller and approaches zero as r<1 For a=10 , r=−2 10,−20, 40, −80, 160,… The terms alternate in sign and grow in magnitude because ∣r∣>1.
Nth Term of Geometric Progression
The nth term of a geometric progression (GP) is the term at a specific position in the sequence, calculated based on the first term and the common ratio. Here's the explanation step-by-step:
- The first term of the sequence is denoted as a (or a 1 ).
- The second term a 2 is obtained by multiplying the first term by the common ratio r: a 2 =a×r.
- The third term, a 3 , is obtained by multiplying the second term by r: a 3 = a×r 2
Following this pattern, the general form for the n-th term is a n = a × r n−1 , where:
- a is the first term
- r is the common ratio
- n is the position of the term.
Let's understand the concept of the nth term of a geometric progression more clearly with an example below. If a=2, r=3, and n=4, find the 4th term: a 4 = a× r n − 1 = 2×3 4−1 = 2 × 3 3 =2 × 27 = 54 So, the 4th term is 54 . Nth Term from the Last: If you need the nth term starting from the last term , use the formula: a n =l/r n−1 where:
- l is the last term,
- r is the common ratio,
- n is the position from the last.
Nth Term from the Last Example: If l = 162, r = 3, and n = 2 a n = l/r n−1 =162/3 2−1 = 162/3 = 54 So, the 2nd term from the last is 54 . This framework allows you to find any term in a geometric sequence, either from the beginning or the end.
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Sum of N term of Geometric Progression
The sum of the first n terms of a geometric progression (GP) depends on the common ratio r .
For a finite GP (with n terms):
The sum is given by: ![um of N term of Geometric Progression]()
Where:
- a = first term
- r = common ratio
- n = number of terms
Example:
For a GP 2,6,18,54: a = 2, r = 3, n = 4 Sum = 80
For an infinite GP (when n→∞ and ∣r∣<1).
The sum is given by: S ∞ =a/1−r
Example:
For a GP 1, 0.5, 0.25, 0.125 a=1, r=0.5 S ∞ =1/1−0.5 = 2 Sum = 2
Geometric Progression Examples in Real Life
Here are a few geometric progression examples:
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Doubling candies every day: 1, 2, 4, 8, 16...
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Folding a paper: Each fold doubles the layers.
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Investment growth: Money multiplies over time.
These show how GP in maths is not just for school it’s everywhere!
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Recursive Formula for a Geometric Sequence
A recursive formula defines each term of a sequence in terms of the previous term. For a geometric sequence, the recursive formula is: term(n)=term(n−1)×r Where:
- term (n) is the current term,
- term (n−1) is the previous term,
- r is the common ratio,
- n≥2
The first term ( a 1 ) is given separately to start the sequence. Example: Recursive Formula for 1, 2, 4, 8, 16, 32, The first term is a 1 = 1 Find the common ratio ( r) by dividing the second term by the first term: r=term(2)term(1) Recursive formula: term(n)=term(n−1)×2 This means:
- term(2)=term(1)×2=1×2=2
- term(3)=term(2)×2=2×2=4
- term(4)=term(3)×2=4×2=8 and so on.
Solved Examples on Geometric Progression
Here are more examples to help understand the concept of geometric progression better:
Example 1: Find the first six terms of a GP if the first term is 3 and the common ratio is 2.
Solution:
Given: a=3, r=2a = 3, r = 2 a = 3 , r = 2 The terms of the GP are: a, ar, ar 2 , ar 3 , ar 4 , ar 5
- a=3
- ar=3×2=6
- ar 2 =3×2 2 =3×4=12
- ar 3 =3×2 3 =3×8=24
- ar 4 =3×2 4 =3×16=48
- ar
5 =3×2 5 =3×32=96
The first six terms are: 3, 6, 12, 24, 48, 96
Example 2: Calculate the sum of the infinite GP 1, 12, 14, 18,...
Solution:
For an infinite GP, the sum is given by: S∞=a/1−r,if ∣r∣< 1 Here, a=1 , r=1/2 S∞=1/1−1/2=1/12=2 The sum of the infinite GP is 2 .
Example 3: Verify if the sequence 4, 12, 36, 108,... is a GP and find its common ratio.
Solution:
To check if the sequence is a GP, divide consecutive terms: 12/4=3, 36/12=3,108/36=3 Since the ratio is constant ( r = 3) , the sequence is a GP with a common ratio of 3 .
Difference Between Geometric and Arithmetic Progression
A Geometric Progression (GP) is a sequence where each term is obtained by multiplying the previous term by a fixed number called the common ratio. In contrast, an Arithmetic Progression (AP) is a sequence where each term is found by adding a fixed number called the common difference to the previous term.
For example, in GP (2, 6, 18, 54, 162), each term is multiplied by 3 (common ratio = 3), leading to rapid growth. In AP (3, 7, 11, 15, 19), each term increases by 4 (common difference = 4), resulting in a steady, linear increase.
GP grows exponentially, meaning the numbers can increase or decrease much faster than AP, which changes at a constant rate. The general formula for the nth term of GP is a n =a 1 ×r (n−1) , while for AP, it is a n =a 1 +(n−1)×d.
Patterns and Magic in GPs
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GPs grow fast like magic!
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Pick every 2nd number in 2, 4, 8, 16, 32 → you get 2, 8, 32 (still a GP!)
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Take logs of GP numbers, you get AP
GP in maths is everywhere just look around!
Fun Activity: Create Your Own GP!
Choose any number. Let’s say 5. Pick a multiplier. Say, 3. Now build your GP:
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5
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5 × 3 = 15
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15 × 3 = 45
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45 × 3 = 135...
Congrats! You’ve made your own geometric progression. Join Tuition Classes for Kids Online Now!!
