Answer:
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:
[tex]x+2=xr[/tex]
And our third term will be our first term multiplied by the common ratio r twice. Therefore:
[tex]x+3=xr^2[/tex]
Solve for x. From the second term, we can divide both sides by x:
[tex]\displaystyle r=\frac{x+2}{x}[/tex]
Substitute this into the third equation:
[tex]\displaystyle x+3=x\Big(\frac{x+2}{x}\Big)^2[/tex]
Square:
[tex]\displaystyle x+3 = x\Big( \frac{(x+2)^2}{x^2} \Big)[/tex]
Simplify:
[tex]\displaystyle x+3=\frac{(x+2)^2}{x}[/tex]
We can multiply both sides by x:
[tex]x(x+3)=(x+2)^2[/tex]
Expand:
[tex]x^2+3x=x^2+4x+4[/tex]
Isolate the x:
[tex]-x=4[/tex]
Hence, our first term is:
[tex]x=-4[/tex]
Then our common ratio r is:
[tex]\displaystyle r=\frac{(-4)+2}{-4}=\frac{-2}{-4}=\frac{1}{2}[/tex]
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.
