Mathematical Series: Exploring the Sum of Harmonic Progression
by Yuvi K - December 16, 2023
What is a Mathematical Series?
A mathematical series is a sequence of numbers where each subsequent term is related to the one before it. The terms of the series could be mathematical equations, written operations or geometric shapes. Series can be continuous or finite and both can help us better understand the relationships between numbers, as well as help us predict and analyse patterns in data.
What is Harmonic Progression? (हार्मोनिक प्रगति क्या है?)
Harmonic Progression is a type of mathematical series in which the reciprocals of the terms progress in geometric progression while the numbers increase arithmetically. It is denoted by Hn. This type of arithmetic series can be expressed in the form of a geometric series when each term is the reciprocal of the previous term multiplied by some constant r.
The sum of a harmonic progression is typically symbolised by Sn or Sn·r. It is important to note that the sum of a harmonic progression can be infinite in some cases while in others, such as when the series consists of finite elements, it is finite.
Sum of the Harmonic Progression from 600 to 1000
The sum of the harmonic progression from 600 to 1000 can be determined by calculating the sum of the first (n) terms of the harmonic series. The formula for finding the sum for a finite set of numbers is given below:
Sn = [ n ( r 1 + r n ) ]/2
Where: r1 is the initial term and rn is the last term of the harmonic series, and n is the total number of terms in the harmonics series.
In this case, r1=600 and rn = 1000 and n=401.
So: Sn = [ 401 ( 600 + 1000 ) ]/2
= 1.20 million
Therefore, the sum of the harmonic progression from 600 to 1000 is 1.20 million.
Exploring the Sum of Harmonic Progression from 600 to 1000
To explore the sum of the harmonic progression from 600 to 1000, we need to calculate the first (n) terms of the harmonic series. This can be done by using the following formula:
r n = a r n – 1
where: a is the common ratio of the harmonic sequence, r1 is the initial term and rn is the last term of the harmonic series.
Let us suppose a=2.
In this case, r1=600 and rn = 1000. Therefore:
r 2 = 2*600
= 1200
r 3 = 2*1200
= 2400
r 4 = 2*2400
= 4800
and so on….
Therefore, the harmonic series can be expressed as:
r1=600, r2=1200, r3=2400, r4=4800 and so on…..
The sum of the harmonic series can also be found using the following formula:
Sn = [ n ( r 1 + r n ) ]/2
Where: r1 is the initial term and rn is the last term of the harmonic series, and n is the total number of terms in the harmonics series.
Therefore, to find the sum of the harmonic progression from 600 to 1000, we need to calculate the sum of the first (n) terms of the harmonic series.
In this case, r1=600 and rn = 1000 and n=401.
So: Sn = [ 401 ( 600 + 1000 ) ]/2
= 1.20 million
Therefore, the sum of the harmonic progression from 600 to 1000 is 1.20 million.
Conclusion
In conclusion, the harmonic progression is an important type of mathematical series where the reciprocals of the terms progress in geometric progression while the numbers increase arithmetically. The sum of a harmonic progression can be infinite or finite, depending on the type of series. The sum of the harmonic progression from 600 to 1000 can be calculated using the formula Sn = [ n ( r 1 + r n ) ]/2. The sum of the harmonic progression from 600 to 1000 is 1.20 million.
