Binomial Theorem And Its Simple Applications Question 14
Question: If the sum of odd numbered terms and the sum of even numbered terms in the expansion of $ {{(x+a)}^{n}} $ are A and B respectively, then the value of $ {{(x^{2}-a^{2})}^{n}} $ is
Options:
A) $ A^{2}-B^{2} $
B) $ A^{2}+B^{2} $
C) 4AB
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
- [a] $ {{(x+a)}^{n}}={{,}^{n}}C_0x^{n}+{{,}^{n}}C_1{x^{n-1}}a+{{,}^{n}}C_2{x^{n-2}}a^{2} $
$ +{{,}^{n}}C_3{x^{n-3}}a^{3}+{{,}^{n}}C_4{x^{n-4}}a^{4}+…. $
$ ={{(}^{n}}C_0x^{n}+{{,}^{n}}C_2{x^{n-2}}a^{2}+{{,}^{n}}C_4{x^{n-4}}a^{4}+….)+ $
$ {{(}^{n}}C_1{x^{n-1}}a+{{,}^{n}}C_3{x^{n-3}}a^{3}+{{,}^{n}}C_5{x^{n-5}}a^{5})+…. $
$ =A+B….(1) $ Similarly, $ {{(x-a)}^{n}}=A-B….(2) $ Multiplying eqns. (1) and (2), we get $ {{(x^{2}-a^{2})}^{n}}=A^{2}-B^{2} $
BETA
