Hewlett Packard, commonly known as HP, is a renowned technology company that specializes in developing and manufacturing innovative electronic devices and software solutions. With a rich history dating back to 1939, HP has continuously demonstrated its commitment to technological advancements and providing cutting-edge products to its customers.
What is Harmonic Progression?
In Mathematics, a progression is defined as a series of numbers arranged in a predictable pattern. It is a type of number set that follows specific, definite rules. Harmonic Progression (HP) is one of the three main types of progressions, along with Arithmetic Progression (AP) and Geometric Progression (GP).
A Harmonic Progression is a sequence of real numbers determined by taking the reciprocals of an arithmetic progression that does not contain zero. In other words, each term in a harmonic progression is the harmonic mean of its two neighbors. For example, if we have an arithmetic progression with terms a, b, c, d, ..., the corresponding harmonic progression can be written as 1/a, 1/b, 1/c, 1/d, ...
Harmonic Mean
The harmonic mean is calculated as the reciprocal of the arithmetic mean of the reciprocals. The formula to calculate the harmonic mean is given by:
Harmonic Mean = n / [(1/a) + (1/b) + (1/c) + (1/d) + ...]
Hewlett & packard: pioneers in tech industryWhere a, b, c, d are the values and n is the number of values present.
Harmonic Progression Formula
To solve problems involving harmonic progression, we can use the corresponding arithmetic progression sum formula. The nth term of the harmonic progression is equal to the reciprocal of the nth term of the corresponding arithmetic progression. The formula to find the nth term of the harmonic progression series is given as:
The nth term of the Harmonic Progression (H.P) = 1 / [a + (n-1)d]
Where:
- a is the first term of the arithmetic progression
- d is the common difference
- n is the number of terms in the arithmetic progression
The above formula can also be written as:
Analyzing hewlett-packard (hpe) stock price: trends, factors, and analyst targetsThe nth term of H.P = 1 / (nth term of the corresponding A.P)
Harmonic Progression Sum
If we have a harmonic progression given by 1/a, 1/a+d, 1/a+2d, ..., 1/a+(n-1)d, the formula to find the sum of the first n terms in the harmonic progression is given by:
Sum of n terms, Sn = (1/d) * ln[(2a+(2n-1)d)/(2a-d)]
Where:
- a is the first term of the arithmetic progression
- d is the common difference of the arithmetic progression
- ln represents the natural logarithm
Relation Between AP, GP, and HP
For any two numbers, if the Arithmetic Mean (AM), Geometric Mean (GM), and Harmonic Mean (HM) are the Arithmetic, Geometric, and Harmonic Mean respectively, then the relationship between these three is given by:
Hpe careers: professional growth opportunities at hewlett packard enterpriseGM^2 = AM * HM, where AM, GM, HM are in GP
This relationship holds true for any set of numbers, where the arithmetic mean is greater than or equal to the geometric mean, which is greater than or equal to the harmonic mean.
Harmonic Progression Examples
Let's explore a couple of examples to understand how to apply the concepts of harmonic progression:
Example 1:
Determine the 4th and 8th term of the harmonic progression 6, 4, ..
Solution:
Hp - leading provider of technology products and servicesGiven:
H.P = 6, 4, 3
Now, let's consider the corresponding arithmetic progression:
A.P = 1/6, 1/4, 1/3, ...
From this A.P, we can calculate the common difference (d) as:
Hp: a legacy of innovation in technologyT2 - T1 = T3 - T2 = 1/12 = d
Using the formula for the nth term of an arithmetic progression:
The 4th term of the A.P = (1/6) + (4-1)(1/12) = (1/6) + (3/12) = 5/12
Similarly, the 8th term of the A.P = (1/6) + (8-1)(1/12) = (1/6) + (7/12) = 9/12 = 3/4
Since the harmonic progression is the reciprocal of the arithmetic progression, we can write the values as:
Complete guide to hewlett packard printer cartridgesThe 4th term of the H.P = 1 / (4th term of the A.P) = 1 / (5/12) = 12/5
The 8th term of the H.P = 1 / (8th term of the A.P) = 1 / (3/4) = 4/3
Example 2:
Compute the 16th term of an H.P if the 6th and 11th terms of the H.P are 10 and 18, respectively.
Solution:
Let's express the H.P in terms of an A.P:
Hewlett packard co career reboot program: path to cyber security successThe 6th term of the A.P = a + 5d = 1/10
The 11th term of the A.P = a + 10d = 1/18
Solving these two equations, we can find the values of a and d:
a = 13/90
d = -2/225
Hp tech support: reliable & affordable assistanceTo find the 16th term, we can use the expression:
a + 15d = (13/90) - (2/15) = 1/90
Thus, the 16th term of the H.P = 1 / (16th term of the A.P) = 90
Therefore, the 16th term of the H.P is 90.
Practice Problems on Harmonic Progression
Here are a few practice problems to further enhance your understanding of harmonic progression:
Hp ink cartridges: everything you need to knowThe second and fifth terms of a harmonic progression are 3/14 and 1/Compute the sum of the 6th and 7th terms of the series.
The sum of the reciprocals of the first 11 terms in a harmonic progression series is 1Determine the 6 terms of the harmonic progression series.
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