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Question
If the sum of odd numbered terms and the sum of even numbered terms in the expansion of \[\left( x + a \right)^n\] are A and B respectively, then the value of \[\left( x^2 - a^2 \right)^n\] is
Options
\[A^2 - B^2\]
\[A^2 + B^2\]
4 AB
none of these
Solution
\[A^2 - B^2\]
\[\text{ If A and B denote respectively the sums of odd terms and even terms in the expansion } (x + a )^n \]
\[\text{ Then } , (x + a )^n = A + B . . . \left( 1 \right)\]
\[ (x - a )^n = A - B . . . \left( 2 \right)\]
\[\text{ Multplying both the equations we get} \]
\[ (x + a )^n (x - a )^n = A^2 - B^2 \]
\[ \Rightarrow ( x^2 - a^2 )^n = A^2 - B^2\]
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