If (1 + x)2n = a0 + a1x + a2x2 + ..... + a2nx2n, then

Question

If (1 + x)2n = a0 + a1x + a2x2 + ..... + a2nx2n, then (A) a0 + a2 + a4 + .... = (1/2) (a0 + a1 + a2 + a3 + ....) (B) an+1 < an (C) an-3 = an+3 (D) All of these

Tardigrade Admin · Accepted Answer

a0 + a1 + a2 + ..... = 22n and a0 + a2 + a4 + .... = 22n -1 an = 2nCn = the greatest coefficient, being the middle coefficient an-3 = 2nCn-3 = 2nC2n - (n- 3) = 2nCn+3 = an+3

Q. If $(1 + x)^{2 n} = a_{0} + a_{1} x + a_{2} x^{2} + ..... + a_{2 n} x^{2 n}$ , then

$a_{0} + a_{2} + a_{4} + .... = \frac{1}{2} (a_{0} + a_{1} + a_{2} + a_{3} + ....)$

$a_{n + 1} < a_{n}$

$a_{n - 3} = a_{n + 3}$

All of these

$a_{0} + a_{1} + a_{2} + ..... = 2^{2 n}$ and $a_{0} + a_{2} + a_{4} + .... = 2^{2 n - 1}$
$a_{n} =^{2 n} C_{n} =$ the greatest coefficient, being the middle coefficient
$a_{n - 3} =^{2 n} C_{n - 3} =^{2 n} C_{2 n - (n - 3)} =^{2 n} C_{n + 3} = a_{n + 3}$

Q. If (1+x)2n=a0​+a1​x+a2​x2+.....+a2n​x2n, then

a0​+a2​+a4​+....=21​(a0​+a1​+a2​+a3​+....)

an+1​<an​

an−3​=an+3​