Concept:
• **Harmonic Progression (HP):** The series of numbers where the reciprocals of the terms are in Arithmetic Progression, is called a Harmonic Progression.
• **Arithmetic Progression (AP):** The series of numbers where the difference of any two consecutive terms is the same, is called an Arithmetic Progression.
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If a be the first term, d be the common difference and n be the number of terms of an AP, then the sequence can be written as follows:
a,a+d,a+2d,…,a+(n−1)d
The nth term of the AP is:
an=a+(n−1)d
**Calculation:**
Every term in HP is the reciprocal of the corresponding terms of an AP.
Let a be the first term and d be the common difference of the AP.
According to the question:
Tm=n
⇒a+(m−1)d1=n
⇒a+(m−1)d=n1 … (1)
Tn=m
⇒a+(n−1)d1=m
⇒a+(n−1)d=m1 … (2)
Subtracting equation (2) from equation (1), we get:
(m−n)d=n1−m1
⇒d=mn1 … (3)
Substituting this value of d in any of the equations (1) or (2), we get:
a+(m−1)(mn1)=n1
⇒a=n1−mnm−1
⇒a=mn1 … (4)
The (m+n)th term of the AP will be:
am+n=a+(m+n−1)d
Using the values in equations (3) and (4), we get:
⇒am+n=mn1+(m+n−1)(mn1)
⇒am+n=mnm+n
And the (m+n)th term of the HP will be:
Tm+n=am+n1=m+nmn
