A circuit containing resistance `R_(1)` inductance `L_(1)` and capacitance `C_(1)` connected in series gives resonance at the same frequency v as a second similar combination of `R_(2), L_(2)` and `C_(2)`. If the two circuits are connected in series, shown that the whole circuit will resonate with the same frequency.

Explanation

As the two circuits `(R_(1), L_(1), C_(1))` and `(R_(2), L_(2), C_(2))` resonate at the same requency, therefore
`v = (1)/(2 pi sqrt(L_(1) C_(1))) = (1)/(2 pi sqrt(L_(2) C_(2)))`
`:. L_(1) C_(1) = L_(2) C_(2)` or `L_(1) = (L_(2) C_(2))/(C_(1))`
When the two circuits are connected in series,
`L = L_(1) + L_(2)` and `C = (C_(1) C_(2))/(C_(1) + C_(2))`
`:. LC = (L_(1) + L_(2)) (C_(1) C_(2))/(C_(1) + C_(2))`
`= (C_(1) C_(2))/(C_(1) + C_(2)) [(L_(2) C_(2))/(C_(1)) + L_(2)]` .....Using (i)
`= (C_(1) C_(2) L_(2))/(C_(1) + C_(2)) [(C_(2))/(C_(1)) + 1]`
`LC = (C_(1) C_(2) L_(2))/(C_(1) + C_(2)) ((C_(1) + C_(2)))/(C_(1)) = L_(2) C_(2)`
Hence, resonant frequency of the comination
`v' = (1)/(2 pi sqrt(LC)) = (1)/(2 pi sqrt(L_(2) C_(2))) = v`
Which was to be proved.