In the expansions of $(x + a)^{n}$ , the sum of odd terms is P and sum of even terms is Q, then the value of $(P^{2} - Q^{2})$ will be

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Solution

The correct option is B
$(x + a)^{n} = x^{n} +^{n} C_{1} x^{n - 1} a +^{n} C_{2} x^{n - 2} a_{2} +^{n} C_{3} x^{n - 3} a^{3} + \dots$
$= (x^{n} +^{n} C_{2} x^{n - 2} a^{2} + \dots) + (^{n} C_{1} x^{n - 1} a +^{n} C_{3} x^{n - 3} a^{3} + \dots)$
$= P + Q$
$∴ (x - a)^{n} = P - Q . As the terms are alter.+ve and -ve$
$∴ P^{2} - Q^{2} = (P + Q) (P - Q) = (x + a)^{n} (x - a)^{n}$
$P^{2} - Q^{2} = (x^{2} - a^{2})^{n}$

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

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

Then the correct statement are:

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