Respuesta :
Answer:
The geometric sequence is
36,24,16,[tex]\frac{32}3[/tex],.......
Step-by-step explanation:
The first term of the g.s is 36.
The 4th term of the given sequence is [tex]\frac{32}{3}[/tex].
The [tex]n^{th}[/tex] term of the geometric sequence is
[tex]T_n=ar^{n-1}[/tex]
Where a is the first term of the geometric sequence and r is the geometric sequence.
Then 4th term of the sequence is
[tex]T_4=ar^{4-1}[/tex]
[tex]\Rightarrow \frac{32}3=36r^3[/tex]
[tex]\Rightarrow r^3=\frac{32}{3\times 36}[/tex]
[tex]\Rightarrow r=\sqrt[3]{ \frac{8}{27}}[/tex]
[tex]\Rightarrow r=\frac{2}{3}[/tex]
Then, second term of the sequence [tex]=36\times( \frac 23)^{2-1}[/tex]
[tex]=36\times \frac 23[/tex]
=24
The third term of the sequence is[tex]=36\times( \frac 23)^{3-1}[/tex]
[tex]=36\times( \frac 23)^2[/tex]
=16
The geometric sequence is
36,24,16,[tex]\frac{32}3[/tex],.......
Answer:
[tex]\frac{2}{3}[/tex]
Step-by-step explanation:
Given:
36, ___, ___, 32/3 is a geometric sequence
We need to find common ratio, r.
Solution:
As here is geometric sequence:-
First term = [tex]ar[/tex] = 36
Fourth term = [tex]ar^{4}[/tex] = [tex]\frac{32}{3}[/tex]
[tex]r=?[/tex]
We can write [tex]ar^{4}[/tex] as [tex]ar\times r^{3}[/tex]
[tex]( \ ar=36, \ given)[/tex]
[tex]ar^{4} =\frac{32}{3} \ given[/tex]
[tex]ar\times r^{3}=\frac{32}{3}[/tex]
[tex]36\times r^{3} =\frac{32}{3}[/tex]
Dividing both sides by 36
[tex]\frac{36}{36} \times r^{3} =\frac{32}{3\times36} \\\\ r^{3} =\frac{32}{108} \\Taking\ cube\ root\ both\ sides\\\sqrt[3]{r^{3} } =\sqrt[3]{\frac{32}{108} } \\ \\ r=\sqrt[3]{\frac{8}{27} } \\ \\ r=\sqrt[3]{\frac{2\times2\times2}{3\times3\times3} }\\ \\ r=\frac{2}{3}[/tex]
Thus, common ratio, r is [tex]\frac{2}{3}[/tex]
2nd term = [tex]a r^{2} =ar\times r=36\times\frac{2}{3} =24\\[/tex]
3rd term=[tex]ar^{3} =ar\times r^{2} =36\times (\frac{2}{3} )^{2} =36\times=\frac{4}{9} =16[/tex]
