What is the difference between geometric progression and harmonic progression
Answers
Answer:
Geometric Progression
A geometric progression is a sequence in which each term is derived by multiplying or dividing the preceding term by a fixed number called the common ratio. For example, the sequence 4, -2, 1, - 1/2,.... is a Geometric Progression (GP) for which - 1/2 is the common ratio.
The general form of a GP is a, ar, ar2, ar3 and so on.
The nth term of a GP series is Tn = arn-1, where a = first term and r = common ratio = Tn/Tn-1) .
The formula applied to calculate sum of first n terms of a GP:
When three quantities are in GP, the middle one is called as the geometric mean of the other two. If a, b and c are three quantities in GP and b is the geometric mean of a and c i.e. b =√ac
The sum of infinite terms of a GP series S∞= a/(1-r) where 0< r<1.
If a is the first term, r is the common ratio of a finite G.P. consisting of m terms, then the nth term from the end will be = arm-n.
The nth term from the end of the G.P. with the last term l and common ratio r is l/(r(n-1)) .
Harmonic Progression
A series of terms is known as a HP series when their reciprocals are in arithmetic progression.
Example: 1/a, 1/(a+d), 1/(a+2d), and so on are in HP because a, a + d, a + 2d are in AP.
The nth term of a HP series is Tn =1/ [a + (n -1) d].
In order to solve a problem on Harmonic Progression, one should make the corresponding AP series and then solve the problem.
nth term of H.P. = 1/(nth term of corresponding A.P.)
If three terms a, b, c are in HP, then b =2ac/(a+c).
OR
The geometric mean is similar, but instead of adding, we multiply the numbers and take the nth root. It’s appropriate for numbers that are distributed along a logarithmic scale - that is, when you’re as likely to find a number twice the size as a number half the size.
The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals of the set of numbers. That is, if the numbers are a1,a2,...an then the harmonic mean is n1a1+1a2+...+1an. It turns up in quite a few physics situations, for example it’s the average resistance of two resistors in parallel. (Whereas the average resistance of two resistors in series is the arithmetic mean)
As a general rule, a good way to work out which mean to use is to ask this question: suppose I replace all these values with this value, will I get the same ‘total’? So, for example, if I’m calculating a ‘total’ by multiplying values together, it makes sense to use the geometric mean. If I’m adding them, I should use the