What is r, the ratio between 2 consecutive terms?
second term=1/2
third term=1/4
ratio r=third/second=(1/4)/(1/2)=(1/2)

Using the formula for the sum of the first n terms of a geometric series, what is the sum of the first 10 terms? Please round your answer to 4 decimals.

a=1,n=10,r=0.5
sum for n terms=
=
=
=
=
=1.998046875
=1.9980(rounding off)

Using the formula for the sum of the first n terms of a geometric series, what is the sum of the first 12 terms? Please round your answer to 4 decimals.

a=1,n=12,r=0.5
sum for n terms=
=
=
=
=
=
=1.99951171875
=1.9995(rounding off)

What observation can make about these sums? In particular, what number does it appear that the sum will always be smaller than?
The number will approach 2.
It will reach 1.999999999999999999999999....upto infinity, but it will always be smaller than 2

Hope this helps,
-xC