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ધન પૂર્ણાક સંખ્યા ${n_1},{n_2}$ માટે સમીકરણ ${(1 + i)^{{n_1}}} + {(1 + {i^3})^{{n_1}}} + {(1 + {i^5})^{{n_2}}} + {(1 + {i^7})^{{n_2}}}$ ની કિમત વાસ્તવિક થાય તો . . . . .(કે જ્યાં $i = \sqrt { - 1} $ )
${n_1} = {n_2} + 1$
${n_1} = {n_2} - 1$
${n_1} = {n_2}$
${n_1} > 0,{n_2} > 0$
Solution
(d)Using ${i^3} = – i,{i^5} = i$ and ${i^7} = – i$, we can write the given expression as ${(1 + i)^{{n_1}}} + {(1 – i)^{{n_1}}} + {(1 + i)^{{n_2}}} + {(1 – i)^{{n_2}}}$
$ = 2{[^{{n_1}}}{C_0}{ + ^{{n_1}}}{C_2}{i^2}{ + ^{{n_1}}}{C_4}{i^4} + …..]$
$ + 2{[^{{n_2}}}{C_0}{ + ^{{n_2}}}{C_2}{i^2}{ + ^{{n_2}}}{C_4}{i^4} + …..]$
$ = 2{[^{{n_1}}}{C_0}{ – ^{{n_1}}}{C_2}{ + ^{{n_1}}}{C_4} + ….]$
$ + 2{[^{{n_2}}}{C_0}{ – ^{{n_2}}}{C_2}{ + ^{{n_2}}}{C_4} + ….]$
This is a real number irrespective of the values of ${n_1}$and ${n_2}$.
