Question

If $T_0, T_1, T_2.....T_n$ represent the terms in the expansion of $ (x + a)^n$, then $(T_0 -T_2 + T_4 - .......)^2 + (T_1 - T_3 + T_5 - .....)^2 =$

• A. $(x^2 + a^2 )$
• B. $(x^2 + a^2 )^n$
• C. $(x^2 + a^2 )^{1/n}$
• D. $(x^2 + a^2 )^{-1/n}$

Answer 1

The Correct Option is B

From the given condition, replacing a by $ai$
and $- ai$ respectively, we get
$\left(x +ai\right)^{n} = \left(T_{0} - T_{2} + T_{4} - ......\right) + i\left(T_{1} - T_{3 } + T_{5} - .....\right) $ .....(i)
and $ \left(x -ai\right)^{n} =\left(T_{0} -T_{2} +T_{4} - .....\right)-i \left(T_{1} - T_{3} +T_{5} - ....\right)$ ....(ii)
Multiplying (ii) and (i) we get required result i.e.,
$ \left(x^{2} +a^{2}\right)^{n} = \left(T_{0} -T_{2} +T_{4} -....\right)^{2} + \left(T_{1} -T_{3} +T_{5} -....\right)^{2} $