The sum of the first two terms of a geometric sequence is 3. The sum of the next two terms is 4/3. Find the first four terms of the sequence.

Let the four terms be a,b,c,d, and be the common ratio.

b=ar
c=ar²
d=ar³

a+b=a(1+r)=3
c+d=ar²(1+r)=4/3

(c+d)/(a+b) = r² = (4/3)/3=4/9
=>
r=√(4/9)=±2/3, assume 2/3

a+b=a(1+r)=a(1+2/3)=5a/3=3
=>
a=9/5
b=ar=6/5
c=ar²=4/5
d=ar³=8/15

Thanks a lot:)

To find the first four terms of the geometric sequence, we need to find the common ratio and the first term.

Let's represent the first term of the sequence as 'a' and the common ratio as 'r'.

From the given information, the sum of the first two terms of the sequence is 3, which gives us the equation:
a + ar = 3 .....(1)

Similarly, the sum of the next two terms of the sequence is 4/3, giving us the equation:
ar^2 + ar^3 = 4/3 .....(2)

We now have a system of two equations with two unknowns. We can solve this system to find the values of 'a' and 'r.'

To solve the system, we will use substitution and eliminate one of the variables. Let's solve equation (1) for 'a' and substitute it into equation (2):

From equation (1), we have a = 3 - ar.

Now, substitute this expression for 'a' in equation (2):

(3 - ar)r^2 + (3 - ar)r^3 = 4/3.

Expand the equation:
3r^2 - ar^3 + 3r^3 - ar^4 = 4/3.

Combine like terms:
3r^2 + 3r^3 - ar^3 - ar^4 = 4/3.

We can now solve this equation to find the possible values of 'r.'

Once we find the values of 'r', substitute them back into equation (1) to find the corresponding values of 'a'.

Finally, the first four terms of the sequence are given by 'a', 'ar', 'ar^2', and 'ar^3'.