Binomial Theorem And Its Simple Applications Question 100
Question: Sum of odd terms is A and sum of even terms is B in the expansion $ {{(x+a)}^{n}}, $ then
[RPET 1987; UPSEAT 2004]
Options:
A) $ AB=\frac{1}{4}{{(x-a)}^{2n}}-{{(x+a)}^{2n}} $
B) $ 2AB={{(x+a)}^{2n}}-{{(x-a)}^{2n}} $
C) $ 4AB={{(x+a)}^{2n}}-{{(x-a)}^{2n}} $
D) None of these
Show Answer
Answer:
Correct Answer: C
Solution:
- $ {{(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}+….. $
But by the condition, $ A={{,}^{n}}C_0x^{n}+{{,}^{n}}C_2{x^{n-2}}a^{2}+{{,}^{n}}C_4{x^{n-4}}a^{4}+…… $ and $ B={{,}^{n}}C_1{x^{n-1}}a+{{,}^{n}}C_3{x^{n-3}}a^{3}+…… $
Hence $ AB=\frac{1}{4}{ {{(x+a)}^{2n}}-{{(x-a)}^{2n}} } $ or $ 4AB={{(x+a)}^{2n}}-{{(x-a)}^{2n}} $
BETA
