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Geometric Progression
A progression is an important concept in mathematics and statistics. It is defined by a series that advances logically. Arithmetic progression (AP), Harmonic Progression (HP) and Geometric Progression (GP) are the 3 types of progression used in mathematics. This article discusses Geometric progression, geometric sequence, and Infinite geometric progression applications in the real world.
What is Geometric Progression (GP)?
Geometric progression is a simple concept that states that In a series, the value succeeding the first value is made of the previous value multiplied with a constant. Hence, the ratio in the series is called the common ratio. In simple terms, the numbers ahead in a geometric progression is a result of the preceding number x common ratio.
For example – In a geometric series 1, 3, 9, 27, 81, 243,
If we follow closely, the 27 is a result of 9 x 3 (constant in the series)
Similarly, the numbers 3 and 81 are also a result of their previous numbers multiplying with their constant.
Now, taking a ratio of 2 consecutive terms in the series
a1 = 1
a2 = 3
a3 = 9
a4 = 27
Now, a2a1 = 31 = 3
a3a2 = 93 = 3
a4a3 = 279 = 3
Hence, we can say that a2a1= a3a2= a4a3= anan-1
In addition, the geometric progression is a sequence in which the initial term is non zero, and each subsequent term is obtained by multiplying the previous term by a set quantity.
Sum of Geometric Progression
A geometric series is a sum of an infinite number of terms with a constant ratio between the terms. There are two types of sum of Geometric Progression.
Sum of Finite Geometric Progression
If the number of terms in the GP series is limited, It is called finite Geometric progression.
For sum of ‘n’ terms
Sn = a(rn-1)r -1 where r1
Sn = Sum of ‘n’ terms
a = first term
r = ratio of the GP
Sum of Infinite Geometric Progression
The GP is called infinite GP if the number of terms in it is not limited. The formula for finding the sum of a given GP to infinity is
S = n=1arn-1 = a1-r; -1 < r < 1
Here,
S∞ = Sum of infinite geometric progression
a = First term of G.P.
r = Common ratio of G.P.
n = Number of terms
Geometric Sequence Formula
The sequence is a set of things that are in order. Additionally, Each term can be calculated by multiplying the previous term by a constant in a Geometric Sequence. In general terms, A geometric sequence is written as –
(a, ar, ar2, ar3, …)
Where a is the first term while r is the constant. It is important to note that the value of r should not be equal to zero.
Hence, if the r = 0 then the sequence is not geometric.
To sum up geometric sequence,
a + ar + ar2 + … + ar(n-1)
(Each term is ark, where k starts at 0 and goes up to n-1)
Applications of Infinite Geometric Progression
Recurring or Periodic Decimals
Evaluating actual-world recurring or periodic decimals is an intriguing application of a geometric progression with infinitely many terms.
When we try to express a common fraction in decimal forms, such as 3/8 or 4/11, the decimal invariably terminates or repeats in blocks. As a result, 38 =0.375 (decimal ends) and 4/11=0.363636… (repeats in decimal)
The remainders in the division procedure used to express the fraction p/q as a decimal fraction can only be 0,1,2,3,4,…,q–1. If we get a remainder of 0 at any point throughout the division, the procedure ends. Otherwise, one of the remainders 0,1,2,3,4,…,q–1 must recur after no more than q divisors, and the decimal begins to repeat.
Infinite Geometric progression examples
Example 1
Find the sum of the infinite geometric sequence
2, 43, 89, 1627, ….
Solution
a1 = 2, r = 23< 1 then,
S = a11-r= 21-23 = 6
Example 2
200 kg of cement is transported to the market by a truck. Every day, the amount of cement transported by the truck grows by 15%. What is the total amount of cement transported to the market by the truck in a week? The answer should be rounded to the nearest whole number.
Solution:
The initial cement quantity is a = 200 kg.
The common ratio is r = 1.15 since the quantity grows by 15% every day.
n = 7 is the number of days in a week.
The sum of n terms of a GP is used to calculate the total amount of cement transported in a week.
S =a(rn-1)r-1
S = 200(1.157-1)1.15 -1
= 1334 kg
Therefore, the total quantity of cement is 1334 kg.
Example 3
An infinite G. P. with favourable terms has a sum of 48, while its first two terms have 36. Find the second term.
Let ‘a’ be the first term and ‘r’ be the common ratio of the G.P.
We have a1-r= 48 ⇒ a = 48(1-r)…(i)
Also it is given that a +ar = 36
⇒ a(1+r) = 36
⇒ 48(1-r) (1 + r) = 36 (from (i))
⇒ 1 – r2 = 34⇒ r2= 14⇒ r = 12
when r = 12, (i) ⇒ a = 48 x12 = 24 and the second term = ar = 24 x 12=12
When r = -12, the terms of the G.P. will become negative.
So the second term is 12.
Conclusion
Infinite Geometric progressions are an essential topic in mathematics and are used widely in science, finance, and research. Furthermore, It is used where the data is infinite or is constantly changing. The applications should be learned well for actual-world usage.
