Step-by-step explanation:
0.25 = 1/4 = 2^-2
128 ≈ 2⁷ = a1
2^-2 = 2⁷/2⁹
therefore,
0.25 = a9
the 9th term.
Answer:
0.25 is the 10th term.
Step-by-step explanation:
First, let's determine whether it is an Arithmetic Sequence or a Geometric Sequence:
Given:
Since U₂-U₁ ≠ U₃-U₂, but U₂÷U₁ = U₃÷U₂, therefore it is a Geometric Sequence with formula:
[tex]\boxed{U_n=U_1\cdot r^{(n-1)}}[/tex]
Now, we are going to find the ratio (r)
[tex]U_2=U_1\cdot r^{(2-1)}[/tex]
[tex]64=128\cdot r[/tex]
[tex]\displaystyle r=\frac{64}{128}[/tex]
[tex]\bf \displaystyle r=\frac{1}{2}[/tex]
The n-term of 0.25:
[tex]U_n=U_1\cdot r^{(n-1)}[/tex]
[tex]0.25=128\cdot \frac{1}{2} ^{(n-1)}[/tex]
[tex]\displaystyle\frac{0.25}{128} =\frac{1}{2} ^{(n-1)}[/tex]
[tex]\displaystyle\frac{0.25}{128}\times\frac{4}{4} =\frac{1}{2} ^{(n-1)}[/tex]
[tex]\displaystyle\frac{1}{512} =\frac{1}{2} ^{(n-1)}[/tex]
[tex]\displaystyle\frac{1}{2^9} =\frac{1}{2} ^{(n-1)}[/tex]
[tex]\displaystyle\left(\frac{1}{2}\right)^{9} =\frac{1}{2} ^{(n-1)}[/tex]
[tex]9=n-1[/tex]
[tex]\bf n=10[/tex]
