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If A is the sum of the odd terms and B the sum of even terms i

Question

If $A$ is the sum of the odd terms and $B$ the sum of even terms in the expansion of $(x + a)^{n}$ , then $A^{2} - B^{2} =$

(x^{2} + a^{2})^{n}

(x^{2} - a^{2})^{n}

2 (x^{2} - a^{2})^{n}

None of these

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Scarlett Discussed

A

is the sum of the odd terms and

B

the sum of even terms in the expansion of

(x + a)^{n}

, then

A^{2} - B^{2} =

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10 mins ago

Text solutionVerified

We have,

(x + a)^{n} =^{n} C_{0} x^{n} +^{n} C_{1} x^{n - 1} a^{1} +^{n} C_{2} x^{n - 2} a^{2} +^{n} C_{3} x^{n - 3} a^{3} + ... +^{n} C_{n} a^{n} = (^{n} C_{0} x^{n} +^{n} C_{2} x^{n - 2} a^{2} + ...) + (^{n} C_{1} x^{n - 1} a^{1} +^{n} C_{3} x^{n - 3} a^{3} + ...) = A + B (x - a)^{n} =^{n} C_{0} x^{n} -^{n} C_{1} x^{n - 1} a^{1} +^{n} C_{2} x^{n - 2} a^{2} -^{n} C_{3} x^{n - 3} a^{3} = (^{n} C_{0} x^{n} +^{n} C_{2} x^{n - 2} a^{2} + ...) - (^{n} C_{1} x^{n - 1} a^{1} +^{n} C_{3} x^{n - 3} a^{3} + ...) = A - B ∴ A^{2} - B^{2} = (A + B) (A - B) = (x + a)^{n} (x - a)^{n} = (x^{2} - a^{2})^{n}

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Question Text	If $A$ is the sum of the odd terms and $B$ the sum of even terms in the expansion of $(x + a)^{n}$ , then $A^{2} - B^{2} =$
Answer Type	Text solution:1
Upvotes	150