Geometric Progression (GP) | Formula and Properties
Geometric Progression (GP) is a sequence of numbers where each next term in the progression is produced by multiplying the previous term by a fixed number. The fixed number is called the Common Ratio.
Â Let’s learn the formulas and properties of Geometric Progression with the help of solved examples.
What is Geometric Sequence?
Geometric sequence is a series of numbers in which the ratio between two consecutive terms is constant. This ratio is known as the common ratio denoted by ‘r’, where r â‰ 0.
The n^{th }term of Geometric series is denoted by a_{n }and the elements of the sequence are written as a_{1}, a_{2}, a_{3,} a_{4}, …, a_{n.}
Condition for the given sequence to b a geometric sequence :
a_{2}/a_{1} = a_{3}/a_{2} = Â … = a_{n}/a_{n-1 }= r (common ratio)
Geometric Progression Formula
The list of formulas related to GP is given below :
Â | Formulas | Â |
---|---|---|
General Form | a,ar,ar^{2},ar^{3},â€¦ | a is the first term, r is the common ratio. |
nth Term of a GP | T_{n} = ar^{n-1} | T_{n} is the nth term, a is the first term, r is the common ratio. |
Common Ratio | r = T_{n}/ T_{n-1}â€‹â€‹ | T_{n}â€‹ and T_{n-1}â€‹ are consecutive terms of the GP. |
Sum of First n Terms (r > 1) | S_{n} = a[(r^{n} â€“ 1)/(r â€“ 1)] | S_{n}â€‹ is the sum of first n terms, and >1r>1. |
Sum of First n Terms (r < 1) | S_{n} = a[(1 â€“ r^{n})/(1 â€“ r)] | S_{n} is the sum of first n terms, and <1r<1. |
nth Term from End (finite GP) | r = l/ [r(n â€“ 1)] | l is the last term, n is the term position from the end. |
Sum of Infinite GP | Valid only if 0 < r < 1. | S_{âˆž}= a/(1 â€“ r) â€‹ |
Geometric Mean | b=acâ€‹ | For three quantities a,b,c in GP, b is the geometric mean of a andc. |
kth Term from End (finite GP) | T_{k} = ar^{n-k}. | T_{k} is the kth term from the end, n is the total number of terms. |
General Form of Geometric Progression
The given sequence can also be written as:
a, ar, ar^{2}, ar^{3}, … , ar^{n-1 Â }
Here, r is the common ratio and a is the scale factor
Common ratio of Geometric Series is given by:
r = successive term/preceding term = ar^{n-1 }/ ar^{n-2}
Nth Term of Geometric Progression
The terms of a GP are represented as a_{1}, a_{2}, a_{3}, a_{4}, â€¦, a_{n.}
Expressing all these terms according to the first term a_{1}, we get
a_{1 }= a
a_{2 }= a_{1}r
a_{3 }= a_{2}r = (a_{1}r)r = a_{1}r^{2}
a_{4 }= a_{3}r = (a_{1}r^{2})r = a_{1}r^{3}
â€¦
a_{m }= a_{1}r^{mâˆ’1}
â€¦
Similarly,
a_{n }= a_{1}r^{n – 1}
General term or nth term of a Geometric Sequence a, ar, ar^{2}, ar^{3}, ar^{4} is given by :Â
a_{n} = ar^{n-1}
where,Â
a_{1} = first term,Â
a_{2} = 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
Sum of N Terms of GP
The sum of geometric progression is given by
S = a_{1 }+ a_{2 }+ a_{3 }+ â€¦ + a_{n}
S = a_{1 }+ a_{1}r + a_{1}r^{2 }+ a_{1}r^{3 }+ â€¦ + a_{1}r^{nâˆ’1} Â Â ….equation (1)
Multiply both sides of Equation (1) by r (common ratio), and we get
S Ã— r= a_{1}r + a_{1}r^{2 }+^{ }a_{1}r^{3 }+ a_{1}r^{4 }+ â€¦ + a_{1}r^{n} Â Â ….equation (2)
Subtract Equation (2) from Equation (1)
S – Sr = a_{1} – a_{1}r^{n}
(1 – r)S = a_{1}(1 – r^{n})
S_{n} = a_{1}(1 – r^{n})/(1 – r), if r<1
Now, Subtracting Equation (1) from Equation (2) will give
Sr – S = a_{1}r^{n }–^{ }a_{1}
(r – 1)S = a_{1}(r^{n}-1)
Hence,Â Sum of First n Terms of GP is given by:
S_{n} = a(1 – r^{n})/(1 – r), if r < 1
S_{n} = a(r^{n} -1)/(r – 1), if r > 1
Sum of Infinite Geometric Progression
The number of terms in infinite geometric progression will approach infinity (n = âˆž). The sum of infinite geometric progression can only be defined at the range of |r| < 1.
Let us take a geometric sequence a, ar, ar^{2}, … which has infinite terms. S_{âˆž} denotes the sum of infinite terms of that sequence, then
S_{âˆž} = a + ar + ar^{2 }+ ar^{3}+ … + ar^{n }+..(1)
Multiply both sides by r,
rS_{âˆž} = ar + ar^{2 }+ ar^{3}+ … … (2)
subtracting eq (2) from eq (1),
S_{âˆž} – rS_{âˆž} = a
S_{âˆž} (1 – r) = a
Thus, Sum of Infinite Geometric Progression is given by,
S_{âˆž}= a/(1-r), where |r| < 1
Properties of Geometric Progression
Geometric Sequence has the following key properties :
- a^{2}_{k} = a_{k-1 }Ã— a_{k+1}
- a_{1} Ã— a_{n} = a_{2} Ã— a_{n-1} =…= a_{k} Ã— a_{n-k+1}
- 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.
- Reciprocal of all the terms in GP also forms a GP.
- If all the terms in a GP are raised to the same power, then the new series is also in GP.
- If y^{2} = xz, then the three non-zero terms x, y, and z are in GP.
Types of Geometric Progression
GP is further classified into two types, which are:
- Finite Geometric Progression (Finite GP)
- 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, ar^{2}, ar^{3},â€¦â€¦ar^{n-1}, ar^{n}.Â
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, ar^{2}, ar^{3},â€¦â€¦ar^{n-1}, ar^{n}……, 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,………
Geometric Sequence Recursive Formula
A recursive formula defines the terms of a sequence in relation to the previous value. As opposed to an explicit formula, which defines it in relation to the term number.
For an example, let’s look at the sequence: 1, 2, 4, 8, 16, 32
Recursive formula of Geometric Series is given by
term(n) = term(n – 1) Ã— 2
In order to find any term, we must know the previous one. Each term is the product of the common ratio and the previous term.
term(n) = term(n – 1) Ã— r
Example: Write a recursive formula for the following geometric sequence: 8, 12, 18, 27, â€¦Â
Solution:Â
The first term is given as 6. The common ratio can be found by dividing the second term by the first term.
r = 12/8 = 1.5
Substitute the common ratio into the recursive formula for geometric sequences and define Â a_{1}
term(n) = term(n – 1) Ã— rÂ
= term(n -1) Ã— 1.5 for n>=2
a_{1 }= 6
Read More :
Geometric Progression vs Arithmetic Progression
Here are the key differences between Geometric Progression and Arithmetic Progression :
Difference between Arithmetic Sequence and Geometric Sequence | ||
---|---|---|
Â | 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, denoted as ‘d’. | The common ratio, denoted as ‘r’. |
General Formula | The nth term is given by anâ€‹=a_{1}+(nâˆ’1)d, where a_{1}â€‹ is the first term and ‘d’ is the common difference. | The nth term is given by, anâ€‹=a_{1}Ã—r(nâˆ’1), where a_{1}â€‹ 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 S_{n}â€‹= nâ€‹/2[2a_{1} +(nâˆ’1)d] | Given by S_{n} = a_{1}(r^{n} -1)/(r – 1) |
Related :
- Difference between Arithmetic and Geometric Sequences
- Sequences and Series in Maths
- Special Series
- Sequences and Series Formulas
- Arithmetic Progression
Solved Examples on Geometric Sequence
Let’s solve some example problems on Geometric sequence.
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?
Solution:Â
First term, a = 4
Common ratio, r = 5
Now, the first five term of GP is
a, ar, ar^{2}, ar^{3}, ar^{4}
a = 4
ar = 4 Ã— 5 = 20
ar^{2} = 4 Ã— 25 = 100
ar^{3} = 4 Ã— 125 = 500
ar^{4} = 4 Ã— 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: Find the sum of GP: 1, 2, 4, 8, and 16.
Solution:Â
Given GP is 1, 2, 4, 8 and 16
First term, a = 1
Common ratio, r = 2/1 = 2 > 1
Number of terms, n = 5
Sum of GP is given by;
S_{n} = a[(r^{n} â€“ 1)/(r â€“ 1)]
S_{5} = 1[(2^{5} â€“ 1)/(2 â€“ 1)]
Â Â Â = 1[(32 â€“ 1)/1]
Â Â Â = 1[31/2]
Â Â Â = 1 Ã— 15.5
Â Â Â = 15.5
Example 3: If 3, 9, 27,â€¦., is the GP, then find its 9th term.
Solution:Â
nth term of GP is given by:
a_{n} = ar^{n-1}
given, GP 3, 9, 27,â€¦.
Here, a = 3 and r = 9/3 = 3
Therefore,
a_{9} = 3 x 3^{9 â€“ 1}
Â Â = 3 Ã— 6561
Â Â = 19683
Geometric Progression- FAQs
What is Geometric Progression Definition?
Geometric Progression (GP) is a specific type of sequence where each succeeding term is produced by multiplying each preceding term by a fixed constant, which is termed a common ratio(r).
For example, 1, 3, 9, 27, 81, …….
What is the common ratio of GP?
Common multiple between each successive term in a GP is termed the common ratio. It is a constant that is multiplied by each term to get the next term in the GP. If a is the first term and ar is the next term, then the common ratio is equal to:
ar/a = r
Write the general form of GP.
General form of a Geometric Progression (GP) is a, ar, ar^{2}, ar^{3}, ar^{4},â€¦,ar^{n-1}
a = First term
r = common ratio
ar^{n-1} = nth term
What is sum of n terms of GP formula?
The formula to find the sum of GP is:
Sn = a + ar + ar^{2} + ar^{3} +â€¦+ ar^{n-1}
Sn = a[(r^{n} â€“ 1)/(r â€“ 1)]
where r â‰ 1 and r > 1
What is Geometric Progression sum to infinity ?
Formula for the sum to infinity of a geometric series is:
S_{âˆž} = a / (1 – r)
where:
- S_{âˆž}â€‹ is the sum to infinity.
- a is the first term of the series.
- r is the common ratio of the series.