The formula for calculating the sum of the terme in a geometric sequence is expressed as
Sn = a1(1 - r^n)/(1 - r)
where
Sn is the sum of the first n terms
a1 is the first term
r is the commin ration
n is the number of terms
From the information given,
a1 = 1/2
when n = 2, S2 = 3/5
Thus,
3/5 = 1/2(1 - r^2)/(1 - r)
Cross multiply. It becomes
3/5 x 2 = (1 - r^2)/(1 - r)
(1 - r^2) can be expressed as (1 - r)(1 + r)
Thus, the expression becomes
6/5 = (1 - r)(1 + r)/(1 - r)
6/5 = 1 + r
r = 6/5 - 1
r = 1/5
Next, we would calculate the sum of the first 10 terms, S10
In this case,
n = 10
S10 = 1/2(1 - (1/5)^10)/(1 - 1/5)
S10 = 1/2(1 - 1/25)/(4/5)
S10 = 0.625
the sum of the first 10 terms of the sequence is 0.625