One useful technique is to convert the HP into an AP of reciprocals. This can simplify calculations, as working with an AP might be more familiar and straightforward.
Example 1: Find the Middle term in this Harmonic series 4, a, 6
(a) 22/5
(b) 21/5
(c) 23/5
(d) 24/5
Ans: (d)
We know that,
we know Inverse of HP is AP.
Example 2: Find the 8th term in the series 1/2, 1/4, 1/6 …….
(a) 1/20
(b) 1/14
(c) 1/16
(d) 1/18
Ans: (c)
We know that,
Convert the HP series in AP
We get 2, 4, 6,……
In the given series,
a (first term) = 2
d (common difference) = 2 …. (4 – 2)
Therefore, 8^{th} term = t_{8} = a + (n  1) d
t_{8} = 2 + (8– 1) 2
t_{8} = 2 + 7 x 2
t_{8} = 2 + 14
t_{8} = 16
Example 1: Find the correct option for A, G, H for the elements 4 and 6 . where A is Arithmetic Mean, G is Geometric Mean & H is Harmonic Mean.
(a) A ≥ G ≥ H
(b) A < G > H
(c) A < G ≥ H
(d) A < G < H
Ans: (a)
Arithmetic mean
Geometric mean (G) = ( 4 × 6 )^{1/2} = 4.89
Hence it is shown that A ≥ G ≥ H
Example 2: If Harmonic Mean of two numbers is 8 and one of the number is 12 then Find Another Number.
(a) 2
(b) 4
(c) 6
(d) 8
Ans: (c)
We know that,
Let another number = b
96 + 8b = 24b
Another number = 6
Example 3: Find the harmonic mean (HM) of 8, 9, 6?
(a) 4.55
(b) 6.65
(c) 7.56
(d) 7.44
Ans: (d)
We know that,
HM = 7.44
314 videos170 docs185 tests

1. What is a harmonic progression? 
2. How can harmonic progressions be useful in mathematics? 
3. What are some common examples of harmonic progressions? 
4. How can one determine the sum of a harmonic progression? 
5. Are there any shortcuts or tricks for working with harmonic progressions? 
314 videos170 docs185 tests


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