Find x so that x, X +2, X'+3 are the first three terms of a geometric sequence. Then find the 5" term of the sequence.

The fifth term is -1/4.

Step-by-step explanation:

We know that the first three terms of the geometric sequence is x, x + 2, and x + 3.

So, our first term is x.

Then our second term will be our first term multiplied by the common ratio r. So:

And our third term will be our first term multiplied by the common ratio r twice. Therefore:

Solve for x. From the second term, we can divide both sides by x:

Substitute this into the third equation:

Square:

Simplify:

We can multiply both sides by x:

Expand:

Isolate the x:

Hence, our first term is:

Then our common ratio r is:

So, our first term is -4 and our common ratio is 1/2.

Then our sequence will be -4, -2, -1, -1/2, -1/4.

You can verify that the first three terms indeed follow the pattern of x, x + 2, and x + 3.

So, our fifth term is -1/4.