Solveeit Logo
Login

Question

Mathematics Question on binomial expansion formula

If T0,T1,T2.....TnT_0, T_1, T_2.....T_n represent the terms in the expansion of (x+a)n (x + a)^n, then (T0−T2+T4−.......)2+(T1−T3+T5−.....)2=(T_0 -T_2 + T_4 - .......)^2 + (T_1 - T_3 + T_5 - .....)^2 =

A

(x2+a2)(x^2 + a^2 )

B

(x2+a2)n(x^2 + a^2 )^n

C

(x2+a2)1/n(x^2 + a^2 )^{1/n}

D

(x2+a2)−1/n(x^2 + a^2 )^{-1/n}

Answer

(x2+a2)n(x^2 + a^2 )^n

Explanation

Solution

From the given condition, replacing a by aiai
and −ai- ai respectively, we get
(x+ai)n=(T0−T2+T4−......)+i(T1−T3+T5−.....)\left(x +ai\right)^{n} = \left(T_{0} - T_{2} + T_{4} - ......\right) + i\left(T_{1} - T_{3 } + T_{5} - .....\right) .....(i)
and (x−ai)n=(T0−T2+T4−.....)−i(T1−T3+T5−....) \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.,
(x2+a2)n=(T0−T2+T4−....)2+(T1−T3+T5−....)2\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}