\({S_1}P - {S_2}P = {({b^2} + {d^2})^{1/2}} - d\)

For distructive interference at \(P\)

\({S_1}P - {S_2}P = \frac{{(2\,n - 1)\lambda }}{2}\)

i.e., \({({b^2} + {d^2})^{1/2}} - d = \frac{{(2n - 1)\lambda }}{2}\)

\( \Rightarrow d\,{\left( {1 + \frac{{{b^2}}}{{{d^2}}}} \right)^{1/2}} - d = \frac{{(2n - 1)\lambda }}{2}\)

\( \Rightarrow d\,\left( {1 + \frac{{{b^2}}}{{2{d^2}}} + ......} \right) - d = \frac{{(2n - 1)\lambda }}{2}\)

(Binomial Expansion)

\( \Rightarrow \frac{b}{{2d}} = \frac{{(2n - 1)\lambda }}{2} \Rightarrow \lambda = \frac{{{b^2}}}{{(2n - 1)d}}\)

For \(n = 1,\;2............,\;\lambda = \frac{{{b^2}}}{d},\;\frac{{{b^2}}}{{3d}}\)