Question

If $T_0,T_1,\cdots ,T_n$ represent the terms in the expansion of $(x+a{)}^n$, then the value of $(T_0-T_2+T_4-\cdots {)}^2+(T_1-T_3+T_5-\cdots {)}^2$ is

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

Answer 1

The correct option is C $(x^2+a^2{)}^n$

We now that,
$(x+a{)}^n=x^n+C_1{x}^{n-1}a+C_2{x}^{n-2}{a}^2+\cdots =T_0+T_1+T_2+T_3+\cdots =(T_0+T_2+T_4+\cdots )+(T_1+T_3+T_5+\cdots )$

The $1^{st}$ bracket involves even powers of $a$ and
$2^{nd}$ 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 $ai$
$(x+ai{)}^n=(T_0-T_2+T_4-\cdots )+i(T_1-T_3+T_5-\cdots )\cdots (i)$
Now, $a\to -ai$
$(x-ai{)}^n=(T_0-T_2+T_4-\cdots )-i(T_1-T_3+T_5-\cdots )\cdots (ii)$ Multiplying $\left(i\right)$ and $\left(ii\right)$
$(x^2+a^2{)}^n=(T_0-T_2+T_4-\cdots {)}^2+(T_1-T_3+T_5-\cdots {)}^2$