Answer:
The fifth term is -1/4.
Step-by-step explanation:
We know that the first three terms of the geometric sequence is <em>x, x</em> + 2, and <em>x </em>+ 3.
So, our first term is <em>x</em>.
Then our second term will be our first term multiplied by the common ratio <em>r</em>. So:
And our third term will be our first term multiplied by the common ratio <em>r</em> twice. Therefore:
Solve for <em>x</em>. From the second term, we can divide both sides by <em>x: </em>
Substitute this into the third equation:
Square:
Simplify:
We can multiply both sides by <em>x: </em>
Expand:
Isolate the <em>x: </em>
Hence, our first term is:
Then our common ratio <em>r</em> 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 <em>x</em>, <em>x</em> + 2, and <em>x</em> + 3.
So, our fifth term is -1/4.