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7.Binomial Theorem
medium
In the expansion 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
A
${({x^2} + {a^2})^n}$
B
${({x^2} - {a^2})^n}$
C
${(x - a)^{2n}}$
D
${(x + a)^{2n}}$
Solution
(b) ${(x + a)^n} = {x^n} + {\,^n}{C_1}{x^{n – 1}}a{ + ^{}}$…..
$ = ({x^n} + {\,^n}{C_2}{x^{n – 2}}{a^2} + …….$+${(^n}{C_1}{x^{n – 1}}a + {\,^n}{C_3}{x^{n – 3}}{a^3} + …..)$
=$P + Q$
$\therefore $ ${(x – a)^n} = P – Q,$ As the terms are alter.
$\therefore $ ${P^2} – {Q^2} = (P + Q)(P – Q) = {(x + a)^n}{(x – a)^n}$
${P^2} – {Q^2} = {({x^2} – {a^2})^n}$
Standard 11
Mathematics
