# Harmonic Progression (HP)

A sequence

*a*_{1},*a*_{2},*a*_{3}, â€¦,*a*, â€¦. of non-zero numbers is called a harmonic progression or a harmonic sequence, if the sequence , â€¦, , â€¦ is an arithmetic progression._{n}

*n***th term of an HP**The

*n*th term of an HP is the reciprocal of the

*n*th term of the corresponding AP. Thus, if

*a*

_{1},

*a*

_{2},

*a*

_{3}, â€¦,

*a*, is an HP and the common difference of the corresponding AP is

_{n}*d*, i.e.,

*d*= , then the

*n*th term of the HP is given by

*a*= .

_{n}In other words,

*n*th term of an HP is the reciprocal of the*n*th term of the corresponding AP.# Insertion of Harmonic Means

Let

*a*,*b*be two given non-zero numbers. If*n*numbers*H*_{1},*H*_{2}, â€¦,*H*are inserted between_{n}*a*and*b*such that the sequence*a*,*H*_{1},*H*_{2},*H*_{3}, â€¦,*H*,_{n}*b*is an HP, then*H*_{1},*H*_{2}, â€¦,*H*are called_{n}*n*harmonic means between*a*and*b*.Now,

*a*,*H*_{1},*H*_{2}, â€¦,*H*,_{n}*b*are in HP.â‡’ , are in AP

Harmonic means of two given numbers

*a*and*b*is*H*= .# AM, GM, and HM of two positive real numbers

Let

*A*,*G*, and*H*be arithmetic, geometric, and harmonic means of two positive numbers*a*and*b,*Then,*A*= ,

*G*= and

*H*=

These three means possess the following properties:

*A*â‰¥*G*â‰¥*H*.*A*,*G*,*H*form a GP, i.e*.,**G*^{2}=*AH.*- If
*A*and*G*be the AM and GM between two positive numbers, then the numbers are*A*Â± . - The equation having
*a*and*b*as its roots in*x*^{2}â€“ 2*Ax*+*G*^{2}= 0. - If
*A*,*G*,*H*are arithmetic, geometric, and harmonic means of three given numbers*a*,*b*, and*c*, then the equation having*a*,*b*,*c*as its roots is*x*^{3}â€“ 3*Ax*^{2}+*x*â€“*G*^{3}= 0.