Harmonic Progression - Examples (with Solutions), Algebra, Quantitative Aptitude | EduRev Notes

UPSC Prelims Paper 2 CSAT - Quant, Verbal & Decision Making

Quant : Harmonic Progression - Examples (with Solutions), Algebra, Quantitative Aptitude | EduRev Notes

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Harmonic Progression (H.P.)

  1. A series of quantities is said to be in a harmonic progression when their reciprocals are in arithmetic progression.
     
  2. e.g.  Harmonic Progression - Examples (with Solutions), Algebra, Quantitative Aptitude | EduRev Notes  are in HP as their reciprocals
     
  3. 3, 5, 7, …, and a, a + d, a + 2d….…. are in AP.

 

nth term of HP

  1. Find the nth term of the corresponding AP and then take its reciprocal.
     
  2. If the HP be  Harmonic Progression - Examples (with Solutions), Algebra, Quantitative Aptitude | EduRev Notes
     
  3. Then the corresponding AP is a, a + d, a + 2d, ……
     
  4. Tn of the AP is a + (n - 1) d
  5. Tnth of the HP is …… Harmonic Progression - Examples (with Solutions), Algebra, Quantitative Aptitude | EduRev Notes
     
  6. In order to solve a question on HP, one should form the corresponding AP.

 

A comparison between AP and GP

Harmonic Progression - Examples (with Solutions), Algebra, Quantitative Aptitude | EduRev Notes


Arithmetic - Geometric progression
a + (a + d)r + (a + 2d)r+ (a + 3d)r3 + ……………. Is the form of Arithmetic geometric progression (A.G.P).
One part of the series is in Arithmetic progression and other part is a Geometric progression.

Harmonic Progression - Examples (with Solutions), Algebra, Quantitative Aptitude | EduRev Notes


Arithmetic geometric series can be solved as explained in the example below:


Relation between AM, GM and HM:


For two positive numbers a and b
Harmonic Progression - Examples (with Solutions), Algebra, Quantitative Aptitude | EduRev Notes

Harmonic Progression - Examples (with Solutions), Algebra, Quantitative Aptitude | EduRev Notes

Harmonic Progression - Examples (with Solutions), Algebra, Quantitative Aptitude | EduRev Notes


This mean A, G, H are in G.P.
Verifying for numbers 1, 2
Harmonic Progression - Examples (with Solutions), Algebra, Quantitative Aptitude | EduRev Notes


Hence  Harmonic Progression - Examples (with Solutions), Algebra, Quantitative Aptitude | EduRev Notes

Harmonic Progression - Examples (with Solutions), Algebra, Quantitative Aptitude | EduRev Notes

Harmonic Progression - Examples (with Solutions), Algebra, Quantitative Aptitude | EduRev Notes

 

Toolkit

Harmonic Progression - Examples (with Solutions), Algebra, Quantitative Aptitude | EduRev Notes

 


Ex.1 Find the sum of 1 + 2x + 3x2 + 4x3 + … ∞

Sol. The given series in an arithmetic-geometric series whose corresponding A.P. and G.P. are 1, 2, 3, 4,…
and 1, x, x2, x3, … respectively. The common ratio of the G.P. is x. Let S denote the sum of the given
series.
Harmonic Progression - Examples (with Solutions), Algebra, Quantitative Aptitude | EduRev Notes



Ex.2 If the first item of an A.P is 12, and 6th term is 27. What is the sum of first 10 terms?


Sol.

Harmonic Progression - Examples (with Solutions), Algebra, Quantitative Aptitude | EduRev Notes


Ex.3 If the fourth & sixth terms of an A.P are 6.5 and 9.5. What is the 9th term of that A.P?
 

Sol.

Harmonic Progression - Examples (with Solutions), Algebra, Quantitative Aptitude | EduRev Notes


Ex.4 What is the arithmetic mean of first 20 terms of an A.P. whose first term is 5 and 4th term is 20?


Sol.

Harmonic Progression - Examples (with Solutions), Algebra, Quantitative Aptitude | EduRev Notes


Ex.5 The first term of a G.P is half of its fourth term. What is the 12th term of that G.P, if its sixth term
 is 6


Sol.

Harmonic Progression - Examples (with Solutions), Algebra, Quantitative Aptitude | EduRev Notes


Ex.6 If the first and fifth terms of a G.P are 2 and 162. What is the sum of these five terms?

Sol.

Harmonic Progression - Examples (with Solutions), Algebra, Quantitative Aptitude | EduRev Notes


Ex.7 What is the value of r + 3r2 + 5r3 + - - - - -

Sol. 

Harmonic Progression - Examples (with Solutions), Algebra, Quantitative Aptitude | EduRev Notes


Ex.8 The first term of a G.P. 2 and common ratio is 3. If the sum of first n terms of this G.P is greater
 than 243 then the minimum value of ‘n’ is


Sol.

Harmonic Progression - Examples (with Solutions), Algebra, Quantitative Aptitude | EduRev Notes


Ex.9 

Harmonic Progression - Examples (with Solutions), Algebra, Quantitative Aptitude | EduRev Notes

Sol.

Harmonic Progression - Examples (with Solutions), Algebra, Quantitative Aptitude | EduRev Notes


Ex.10 Harmonic Progression - Examples (with Solutions), Algebra, Quantitative Aptitude | EduRev Notes 

Sol. 

Harmonic Progression - Examples (with Solutions), Algebra, Quantitative Aptitude | EduRev Notes

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