JEE Main & Advanced Mathematics Sequence & Series Harmonic Mean

If three or more numbers are in H.P., then the numbers lying between the first and last are called harmonic means (H.M.’s) between them. For example 1, 1/3, 1/5, 1/7, 1/9 are in H.P. So 1/3, 1/5 and 1/7 are three H.M.’s between 1 and 1/9.

Also, if a, H, b are in H.P., then H is called harmonic mean between \[a\] and \[b\].

(1) Insertion of harmonic means

(i) Single H.M. between \[a\] and \[b\]\[=\frac{2ab}{a+b}\].

(ii) H, H.M. of \[n\] non-zero numbers \[{{a}_{1}},\,{{a}_{2}},\,{{a}_{3}},\,....,\,{{a}_{n}}\]  is given by \[\frac{1}{H}=\frac{\frac{1}{{{a}_{1}}}+\frac{1}{{{a}_{2}}}+.....+\frac{1}{{{a}_{n}}}}{n}\].

(iii) Let \[a,\,\,b\] be two given numbers. If \[n\] numbers \[{{H}_{1}},\,{{H}_{2}},\,......\,{{H}_{n}}\] are inserted between \[a\] and \[b\] such that the sequence  \[a,\,{{H}_{1}},\,{{H}_{2}},\,{{H}_{3}},......\,{{H}_{n}},\,b\] is a H.P., then \[{{H}_{1}},\,{{H}_{2}},\,......\,{{H}_{n}}\] are called \[n\] harmonic means between \[a\] and \[b\].

Now, \[a,\,{{H}_{1}},\,{{H}_{2}},\,{{H}_{3}},......\,{{H}_{n}},\,b\] are in H.P.

\[\Rightarrow \] \[\frac{1}{a},\,\frac{1}{{{H}_{1}}},\,\frac{1}{{{H}_{2}}},\,......\frac{1}{{{H}_{n}}},\,\frac{1}{b}\] are in A.P.

Let \[D\] be the common difference of this A.P. Then,

\[\frac{1}{b}={{(n+2)}^{th}}\text{ term }={{T}_{n+2}}\]

\[\frac{1}{b}=\frac{1}{a}+(n+1)\,D\]\[\Rightarrow \]\[D=\frac{a-b}{(n+1)\,ab}\].

Thus, if \[n\] harmonic means are inserted between two given numbers \[a\] and \[b,\] then the common difference of the corresponding A.P. is given by \[D=\frac{a-b}{(n+1)\,ab}\].

Also, \[\frac{1}{{{H}_{1}}}=\frac{1}{a}+D\], \[\frac{1}{{{H}_{2}}}=\frac{1}{a}+2D\],…….,\[\frac{1}{{{H}_{n}}}=\frac{1}{a}+nD\], 

where \[D=\frac{a-b}{(n+1)\,ab}\].