Page 37 - 11-Math-6 Sequences and Series

61.. SQeuqaudenracteiscaEnqduSaetriioenss eeLLeeaarrnn..PPuunnjjaabb

= 1+4= 9.
22

Hence the required H.P. is 2 , 2 , 2 , 2 ,...
1 5 9 13

6.12.1 Harmonic Mean : A number H is said to be the harmonic mean (H.M) between two

numbers a and b if a, H, b are in H.P.

Let a, b be the two numbers and H be their H.M. Then 1 , 1 , 1 are in A.P.
a H b

therefore=, 1 1+1 b+a a+b
a=b =ab
H2 2 2ab

and H = 2ab
a+b

For example, H.M. between 3 and 7 is

2 ×=3× 7 2=× 21 21
3 + 7 10 5

6.12.2 n Harmonic Means between two numbers

H1, H2, H3...., Hn are called n harmonic means (H.Ms) between a and b if a, H1, H2, H3,....Hn,b
are in H.P. If we want to insert n H.Ms. between a and b, we first find n A.Ms. A1 , A2,...., An

between 1 and 1 , then take their reciprocals to get n H.Ms between a and b, that is,
ab

1 , 1 ,..., 1 will, be the required n H.Ms. between a and b.
A1 A2 An
Example 3: Find three harmonic means between 1 and 1 .

5 17

Solution: Let A1, A2, A3 be three A.Ms. between 5 and 17, that is, 5, A1, A2, A3,17 are in A.P.

version: 1.1

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