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Algebraic Expression

If T0,T1,⋯,Tn...

Question

If $T_{0}, T_{1}, \dots, T_{n}$ represent the terms in the expansion of $(x + a)^{n}$ , then the value of $(T_{0} - T_{2} + T_{4} - \dots)^{2} + (T_{1} - T_{3} + T_{5} - \dots)^{2}$ is

A

a^{2 n}

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B

(x + a)^{2 n}

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C

(x^{2} + a^{2})^{n}

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D

a^{2 n}

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Solution

The correct option is C $(x^{2} + a^{2})^{n}$
We now that,
$(x + a)^{n} = x^{n} + C_{1} x^{n - 1} a + C_{2} x^{n - 2} a^{2} + \dots = T_{0} + T_{1} + T_{2} + T_{3} + \dots = (T_{0} + T_{2} + T_{4} + \dots) + (T_{1} + T_{3} + T_{5} + \dots)$

The $1^{s t}$ bracket involves even powers of $a$ and
$2^{n d}$ bracket involves odd powers of $a$ .
We know that,
$i^{2} = - 1, i^{4} = 1, i^{3} = - i, i^{5} = i$
Therefore, replacing $a$ by $a i$
$(x + a i)^{n} = (T_{0} - T_{2} + T_{4} - \dots) + i (T_{1} - T_{3} + T_{5} - \dots) \dots (i)$
Now, $a \to - a i$
$(x - a i)^{n} = (T_{0} - T_{2} + T_{4} - \dots) - i (T_{1} - T_{3} + T_{5} - \dots) \dots (i i)$ Multiplying $(i)$ and $(i i)$
$(x^{2} + a^{2})^{n} = (T_{0} - T_{2} + T_{4} - \dots)^{2} + (T_{1} - T_{3} + T_{5} - \dots)^{2}$

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