Answer:
[tex]\frac{1}{256}[/tex]
Step-by-step explanation:
Geometric sequence means there is a common ratio. All that means is term divided previous term is the same across your sequence.
ONE WAY:
So we are given here that:
[tex]\frac{f(2)}{f(1)}=\frac{1}{2}[/tex] and that the first term which is [tex]f(1)[/tex] is 2.
[tex]\frac{f(2)}{2}=\frac{1}{2}[/tex]
This implies [tex]f(2)=1[/tex] after multiplying both sides by 2 and getting that [tex]f(2)=\frac{1}{2}(2)=\frac{2}{2}=1[/tex].
So you have that
2,1,...
basically you can just multiply by 1/2 to keep generating more terms of the sequence.
Third term would be [tex]f(3)=1(\frac{1}{2})=\frac{1}{2}[/tex].
Fourth term would be [tex]f(4)=\frac{1}{2}(\frac{1}{2})=\frac{1}{4}[/tex].
...keep doing this til you get to the 10th term.
ANOTHER WAY:
Let's make a formula.
[tex]f(n)=ar^{n-1}[/tex]
[tex]a[/tex] is the first term.
[tex]r[/tex] is the common ratio.
And we want to figure out what happens at [tex]n=10[/tex].
Let's plug in our information we have
[tex]a=2[/tex]
[tex]r=\frac{1}{2}[/tex]:
[tex]f(10)=2(\frac{1}{2})^{10-1}[/tex]
Put into calculator or do by hand...
[tex]f(10)=2(\frac{1}{2})^9[/tex]
[tex]f(10)=2(\frac{1^9}{2^9})[/tex]
[tex]f(10)=2(\frac{1}{2^9})[/tex]
[tex]f(10)=\frac{2}{2^9}[/tex]
[tex]f(10)=\frac{2}{2(2^8)}[/tex]
[tex]f(10)=\frac{1}{2^8}[/tex]
Scratch work:
[tex]2^8=2^5 \cdot 2^3=32 \cdot 8=256[/tex].
End scratch work.
The answer is that the tenth term is [tex]\frac{1}{256}[/tex]
