Question

If $T_{0}, T_{1}, T_{2}, \ldots, T_{n}$ represent the terms in the expansion of $(x+a)^{n},$ then $\left(T_{0}=T_{2}+T_{4}-\cdots\right)^{2}+\left(T_{1}-T_{3}+T_{5}-\cdots\right)^{2}=$

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

Answer 1

Answer: (B)

$(x+a)^{n}={ }^{n} C_{0} x^{n}+{ }^{n} C_{1} x^{n-1} a+{ }^{n} C_{2} x^{n-2} a^{2}+{ }^{n} C_{3} x^{n-3} a^{3}+\cdots $
$=T_{0}+T_{1}+T_{2}+T_{3}+\cdots$
In (1) replacing $a$ by $a i$
$(x+a i)^{n}={ }^{n} C_{0} x^{n}+{ }^{n} C_{1} x^{n-1} a i+{ }^{n} C_{2} x^{n-2}(a i)^{2}+{ }^{n} C_{3} x^{n-3}(a i)^{3}+\cdots $
$=\left({ }^{n} C_{0} x^{n}-{ }^{n} C_{2} x^{n-2} a^{2}+{ }^{n} C_{4} x^{n-4} a^{4}-\cdots\right) $
$ +i\left({ }^{n} C_{1} x^{n} a-{ }^{n} C_{3} x^{n-3} a^{3}+{ }^{n} C_{5} x^{n-5} a^{5}-\cdots\right) $
$=\left(T_{0}-T_{2}+T_{4}-\cdots\right)+i\left(T_{1}-T_{3}+T_{5}-\cdots\right) $
Taking modulus on both sides and squaring
$|x+a i|^{2 n}=\mid\left(T_{0}-T_{2}+T_{4}-\cdots\right)+i\left(T_{1}-T_{3}+T_{5}-\cdots\right)$
$\Rightarrow \left(x^{2}+a^{2}\right)^{n}=\left(T_{0}-T_{2}+T_{4}-\cdots\right)^{2}+\left(T_{1}-T_{3}+T_{5}-\cdots\right)^{2}$