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BITSAT 2016
Exam
BITSAT
Year
2016
Subject
Mathematics
Chapter
Binomial Theorem

BITSAT 2016 · Mathematics · Binomial Theorem

If the sum of odd numbered terms and the sum of even numbered terms...

If the sum of odd numbered terms and the sum of even numbered terms in the expansion of (x+a)n(x+a)^{n} are AandA andBrespectively,thenthevalueofrespectively, then the value of\left(x^{2}-\right.a2)n\left.a^{2}\right)^{n} is

Options

  1. AA2−B2\mathrm{A}^{2}-\mathrm{B}^{2}
  2. BA2+B2\mathrm{A}^{2}+\mathrm{B}^{2}
  3. C4AB4 \mathrm{AB}
  4. DNone

Correct answer

A. A2−B2\mathrm{A}^{2}-\mathrm{B}^{2}

Step-by-step solution

(x+a)n=(n0)xn+(n1)xn−1a+(n2)xn−2a2+(n3)xn−3a3+(n4)xn−4a4+….=((n0)xn+(n2)xn−2a2+(n4)xn−4a4+….)++((n1)xn−1a+(n3)xn−3a3+(n5)xn−5a5)+….\begin{array}{r} (x+a)^{n}=\binom{n}{0} x^{n}+\binom{n}{1} x^{n-1} a+\binom{n}{2} x^{n-2} a^{2} \\ +\binom{n}{3} x^{n-3} a^{3}+\binom{n}{4} x^{n-4} a^{4}+\ldots . \\ =\left(\binom{n}{0}x^{n}+\binom{n}{2} x^{n-2} a^{2}+\binom{n}{4} x^{n-4} a^{4}+\ldots .\right)+ \\ +\left(\binom{n}{1}x^{n-1} a+\binom{n}{3} x^{n-3} a^{3}+\binom{n}{5} x^{n-5} a^{5}\right)+\ldots . \end{array}=A+B....(i)Similarly,....(i) Similarly,(x-a)^{n}=A-B....(ii)Multiplyingeqns.(1)and(2),weget....(ii) Multiplying eqns. (1) and (2), we get\left(\mathrmx^{2}-\mathrma^{2}\right)^{\mathrm{n}}=$\mathrm{A}^{2}-\mathrm{B}^{2}
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