Note :

(1)   In Lyman Series, transition happens in n = 1 state
from n = 2, 3, . . . . . $$ \propto $$

(2)   In Balmer Series, transition happens in n = 2 state
from n = 3, 4, . . . . . $$ \propto $$

(3)   In Paschen Series, transition happens in n = 3 state
from n = 4, 5, . . . . . $$ \propto $$

(4)   In Bracktt Series, transition happens in n = 4 state
from n = 5, 6 . . . . . . $$ \propto $$

(5)   In Pfund Series, transition happens in n = 5 state
from n = 6, 7, . . . . $$ \propto $$

We know,

$${1 \over \lambda }$$  =  RZ² $$\left( {{1 \over {n_1^2}} - {1 \over {n_2^2}}} \right)$$

Series limit means transition happens

from n = $$ \propto $$ to n = 1, for Lyman Series.

In series limit for Lyman series,

$${1 \over {{\lambda _L}}}$$ = RZ² $$\left( {{1 \over {{1^2}}} - {1 \over \propto }} \right)$$

$$ \Rightarrow $$$$\,\,\,$$ $${1 \over {{\lambda _L}}}$$ = RZ²

We know,

E = $${{hc} \over \lambda }$$ = h$$\gamma $$

$$ \Rightarrow $$$$\,\,\,$$ $$\gamma $$ = $${c \over \lambda }$$

So, frequency in Lyman Series,

$$\gamma $$_L =  $${c \over {{\lambda _L}}}$$  = c $$ \times $$ RZ²

In Pfund series,

n₂ = $$ \propto $$  and  n₁ = 5

$$\therefore\,\,\,$$ $${1 \over {{\lambda _P}}}$$ = RZ²$$\left( {{1 \over {{5^2}}} - {1 \over {{ \propto ^2}}}} \right)$$

$$ \Rightarrow $$$$\,\,\,$$ $${1 \over {{\lambda _P}}}$$ = $${{R{Z^2}} \over {25}}$$

$$\therefore\,\,\,$$ $${\gamma _P}$$   =   $${c \over {{\lambda _P}}}$$ = c $$ \times $$ $${{R{Z^2}} \over {25}}$$

$$\therefore\,\,\,$$ $$\gamma $$_P = $${{cRZ{}^2} \over {25}}$$ = $${{{\gamma _L}} \over {25}}$$    [as   $$\gamma $$_L = cRZ²]