Question

已知数列{xn}满足x1=3,x2=x1/2,...,xn=1/2(xn-1+xn-2),n=3,4,...,则xn等于

Answer 1

以下用^b表示b次方.

x(n) = (x(n-1) + x(n-2)) /2,两边减x(n-1)得

x(n) - x(n-1) = (x(n-1) - x(n-2)) * (-1/2)

所以{ x(n) - x(n-1) }是以x(2)-x(1)为首项,以-1/2为公比的等比数列

所以x(n) - x(n-1) = (x(2)-x(1)) * (-1/2) = (-3/2) (-1/2)^(n-2) = 3 *(-1/2)^(n-1) ,（当n≥3）.

x(n) = x(2) + 3/4 *[ 1 - (-1/2)^(n-2) ] / ( 1 + 1/2)

=3/2 + 1/2 * [1 - (-1/2)^(n-2)]

=2 + (-1/2)^(n-1)

易验证n=1,2时也成立,

所以通项公式为x(n) = 2 + (-1/2)^(n-1)