If for the harmonic progression,t_{7} = frac{ | vedclass

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(C) In a harmonic progression $(HP)$,the reciprocals of the terms form an arithmetic progression $(AP)$.Let the corresponding $AP$ have the first term

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$\frac{1}{49}$

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(C) In a harmonic progression $(HP)$,the reciprocals of the terms form an arithmetic progression $(AP)$.Let the corresponding $AP$ have the first term $A$ and common difference $D$.Given $t_{7} = \frac{1}{10} \Rightarrow T_{7} = 10$,where $T_{n}$ is the $n^{	ext{th}}$ term of the $AP$.Given $t_{12} = \frac{1}{25} \Rightarrow T_{12} = 25$.Using the formula $T_{n} = A + (n-1)D$:$A + 6D = 10$  (Equation $1$)$A + 11D = 25$ (Equation $2$)Subtracting Equation $1$ from Equation $2$: $5D = 15 \Rightarrow D = 3$.Substituting $D = 3$ into Equation $1$: $A + 6(3) = 10$ $\Rightarrow A + 18 = 10$ $\Rightarrow A = -8$.Now,find the $20^{	ext{th}}$ term of the $AP$: $T_{20} = A + 19D = -8 + 19(3) = -8 + 57 = 49$.Therefore,the $20^{	ext{th}}$ term of the $HP$ is $t_{20} = \frac{1}{T_{20}} = \frac{1}{49}$.

If for the harmonic progression,$t_{7} = \frac{1}{10}$ and $t_{12} = \frac{1}{25}$,then $t_{20} =$

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