1. Recall the hydrogen spectral series

For hydrogen atom:

• Lyman series: transitions ending at $n=1$
• Balmer series: transitions ending at $n=2$
• Paschen series: transitions ending at $n=3$

Also:

• First member means transition from the next higher level.
• Second member means one more level above that.
• Third member means yet another level above.
• Series limit means transition from $n=\infty$ to the lower level of that series.

2. Members of each series

Lyman series

Ends at $n=1$:

• first member: $2 \to 1$
• second member: $3 \to 1$
• third member: $4 \to 1$
• series limit: $\infty \to 1$

Balmer series

Ends at $n=2$:

• first member: $3 \to 2$
• second member: $4 \to 2$
• third member: $5 \to 2$

Paschen series

Ends at $n=3$:

• first member: $4 \to 3$
• second member: $5 \to 3$
• third member: $6 \to 3$

3. Identify transitions A, B and C from the figure pattern

From the standard hydrogen energy-level diagram with marked arrows:

• A is the transition from $n=\infty$ to $n=1$ $Rightarrow$ series limit of Lyman series.
• B is the transition from $n=5$ to $n=2$ $Rightarrow$ third member of Balmer series.
• C is the transition from $n=5$ to $n=3$ $Rightarrow$ second member of Paschen series.

4. Match with options

We need:

• $A$ = series limit of Lyman series
• $B$ = third member of Balmer series
• $C$ = second member of Paschen series

This matches Option C.

5. Check other options briefly

• Option A: says $B$ is second member of Balmer series ($4\to2$), incorrect.
• Option B: says $A$ is ionization potential of hydrogen; that is not the marked transition description here, and $C$ is also third member of Paschen, incorrect.
• Option D: says $A$ is first member of Lyman ($2\to1$), incorrect.

Final Answer

The correct option is C.