Sequence and Series
4.0 Harmonic Sequence or Harmonic Progression (H.P.)
4.0 Harmonic Sequence or Harmonic Progression (H.P.)
A sequence is said to be in harmonic progression if the reciprocal of its terms are in arithmetic progression.
If ${a_1},{a_2},{a_3},...,{a_n}$ is in H.P., then $\frac{1}{{{a_1}}},\frac{1}{{{a_2}}},\frac{1}{{{a_3}}},...,\frac{1}{{{a_n}}}$ is in A.P.
For example, $\frac{1}{2},\frac{1}{4},\frac{1}{6},...$ forms a H.P. because $2,4,6,...$ are in A.P.
- If $a$ and $b$ are the first two terms of an H.P., then $n$th term is $${T_n} = \frac{1}{{\frac{1}{a} + (n - 1)\left( {\frac{1}{b} - \frac{1}{a}} \right)}}$$
- If $a$ and $b$ are the first two terms of an H.P. and $l$ is the last term, then $n$th term from end is $${T'}_n = \frac{1}{{\frac{1}{l} - (n - 1)\left( {\frac{1}{b} - \frac{1}{a}} \right)}}$$
- There is no general formulae for finding out the sum of $n$ terns of H.P. No term of H.P. can be zero.
- Three terms in H.P. can be taken as $$\frac{1}{{a - d}},\frac{1}{a},\frac{1}{{a + d}}$$
- Four terms in H.P. can be taken as $$\frac{1}{{a - 3d}},\frac{1}{{a - d}},\frac{1}{{a + d}},\frac{1}{{a + 3d}}$$
- Five terms in H.P. can be taken as $$\frac{1}{{a - 2d}},\frac{1}{{a - d}},\frac{1}{a},\frac{1}{{a + d}},\frac{1}{{a + 2d}}$$
