The common ratio r=-¼ and the first term a=-2. The series in algebraic terms is a, ar, ar2, ...
-2, 1/2, -1/8, 1/32, -1/128, 1/512, -1/2048, 1/8192, -1/32768, ... Odd terms are negative, even terms are positive.
So there is no term 1/32768. I have to assume there's an error and T9 should be -1/32768, not 1/32768.
Also:
1/32768=arn-1=(-2)(-¼)n-1.
-1/65536=(-¼)n-1; -1/216=(-¼)n-1=(-1/22)n-1; when n is odd, (-¼)n-1=1/22n-2; when n is even, (-¼)n-1=-1/22n-2.
(When n=9, Tn=(-2)(-¼)8=(-2)/65536=-1/32768; when n=10, Tn=(-2)(-¼)9=-(-2)/262144=1/131072.)
1-rn=(1-r)(1+r+r2+r4+...+rn-1), so Sn=a(1+r+r2+r4+...+rn-1)=
a(1-rn)/(1-r); S9=-2(1-(-¼)9)/(1+¼)=-2=-8(1+1/262144)/5
-262145/163840=-52429/32768≅-1.6.