Question

If $(1 + x)^{2n} = a_0 + a_1x + a_2x^2 + ..... + a_{2n}x^{2n}$, then

• A. $a_{0} + a_{2 }+ a_{4} + .... = \frac{1}{2} \left(a_{0 }+ a_{1} + a_{2} + a_{3} + ....\right)$
• B. $a_{n+1} < a_n$
• C. $a_{n-3} = a_{n+3}$
• D. All of these

Answer 1

Answer: (D)

$a_0 + a_1 + a_2 + ..... = 2^{2n}$ and $a_0 + a_2 + a_4 + .... = 2^{2n -1}$
$a_n = ^{2n}C_n =$ the greatest coefficient, being the middle coefficient
$a_{n-3} = ^{2n}C_{n-3} =^{ 2n}C_{2n - \left(n- 3\right)} = ^{2n}C_{n+3} = a_{n+3}$