Question

In the expansion of $(x+a{)}^n$, if the sum of odd terms be P and sum of even terms be Q. Consider the following statements:

(i) $P^2-Q^2=(x^2-a^2{)}^n$

(ii) $P^2+Q^2=(x^2+a^2{)}^n$

(iii) 4PQ = $(x+a{)}^{2n}-(x-a{)}^{2n}$

(iv) P - Q = $(x-a{)}^n$

Then the correct statement are:

• A. (i), (iv)
• B. (i), (iii), (iv)
• C. (i), (iii)
• D. (ii), (iii)

Answer 1

The correct option is B

(i), (iii), (iv)

In the expansion of $(x+a{)}^n$.

P+Q = $(x+a{)}^n$ , P-Q = $(x-a{)}^n$

$\therefore P^2-Q^2=(x^2-a^2{)}^n$

$(P+Q{)}^2-(P-Q{)}^2=4PQ$

$\Rightarrow 4PQ=(x+a{)}^{2n}-(x-a{)}^{2n}$

But $P^2+Q^2\ne (x^2+a^2{)}^n$.

$\therefore \left(i\right),\left(iii\right)and\left(iv\right)$ are correct.