Given that the sum of the first four terms of a geometric sequence is 10 and the fourth term is half the third term . find the common ratio and the first term

a(r^4-1)/(r-1) = 10

T4/T3 = 1/2

the common ratio is the ratio between any term and the previous one.

That should get you started.

To solve this problem, we need to use the properties of a geometric sequence and set up a system of equations.

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

We know that the sum of the first four terms is 10, so we can write the equation:

a + ar + ar^2 + ar^3 = 10 ...(1)

We are also given that the fourth term is half the third term, so we can write another equation:

ar^3 = (1/2) * ar^2 ...(2)

Now, let's solve this system of equations to find the values of 'a' and 'r'.

From equation (2), we can see that ar^3 cancels out on both sides:

(1/2) * ar^2 = ar^2

Dividing both sides of the equation by ar^2, we get:

1/2 = r

So, we have found that the common ratio (r) is 1/2.

Now, substitute the value of 'r' (1/2) in equation (1):

a + a(1/2) + a(1/2)^2 + a(1/2)^3 = 10

Simplifying this equation gives us:

a + (1/2)a + (1/4)a + (1/8)a = 10

Combining the terms on the left side of the equation:

(15/8)a = 10

To solve for 'a', divide both sides of the equation by (15/8):

a = (10 * 8) / 15

Simplifying this gives us:

a = 80/15 = 16/3

Therefore, we have found that the first term (a) is 16/3 and the common ratio (r) is 1/2.