As per given figure A, B and C are the first, second and third...

For ${\lambda}_{1}$

$\dfrac{1}{{\lambda}_{1}} = R{Z}^{2} ( \dfrac{1}{{2}^{2}} - \dfrac{1}{{3}^{2}} ) = R\left(\dfrac{5}{36}\right) . . . ( i )$

For ${\lambda}_{2}$,

$\dfrac{1}{{\lambda}_{2}} = R{Z}^{2} ( \dfrac{1}{{3}^{2}} - \dfrac{1}{{4}^{2}} ) = R\left(\dfrac{7}{144}\right) . . . ( ii )$

Dividing (ii) by (i)

$\dfrac{{\lambda}_{1}}{{\lambda}_{2}} = \dfrac{\dfrac{7}{144}}{\dfrac{5}{36}} = \dfrac{7}{20}$

So,

$\dfrac{7}{4n} = \dfrac{7}{20} ⇒ n = 5$