Respuesta :

Answer:

15th term will be [tex]\frac{25}{2048} }[/tex]

Step-by-step explanation:

In a geometric progression we know any term of the series is represented as [tex]T(n) = a(r)^{n-1}[/tex]

where a = first term, r = common ratio, and n = number of term

Now as per question 4th term is -25 then as pr formula

[tex]T(4) = a (r)^{4-1} = ar^{3} = -25[/tex]------(1)

Now the ninth term is 25/32

[tex]T(9) = ar^{9-1} = ar^{8} = \frac{25}{32}[/tex]-----(2)

Now we put the value of a from equation 1 to equation 2

[tex]a =\frac{-25}{r^{3} }[/tex]

[tex]\frac{-25}{r^{3} } (r^{8})=\frac{25}{32}\\-25r^{5} = \frac{25}{32\\} or -r^{5}=\frac{1}{32}[/tex]

[tex]r^{5} = -\frac{1}{32}[/tex]

[tex]r = -\frac{1}{\sqrt[5]{32}}  = -\frac{1}{2}[/tex]

therefore a = [tex]a(-\frac{1}{2})^{8} =\frac{25}{32}[/tex]

[tex]a=\frac{25}{32}(2)^{8}[/tex]

[tex]a = \frac{25}{2^{5} } (2^{8}) =25(2)^{3} =200[/tex]

Therefore 15th term will be

[tex]200(-\frac{1}{2}) ^{15-1} = 200(\frac{1}{2})^{14}= \frac{25}{2^{11} }= \frac{25}{2048}[/tex]

Answer:c

15th term of the geometric sequence is 0.0122

Step-by-step explanation:

We are given that,

The fourth term and ninth term of the geometric sequence are [tex]-25[/tex] and [tex]\frac{25}{32}[/tex].

Since, the general expression for the nth term of a geometric sequence is [tex]a_{n}=ar^{n-1}[/tex].

So, we get,

[tex]a_{4}=ar^{4-1}=ar^{3}=-25[/tex] and [tex]a_{9}=ar^{9-1}=ar^{8}=\frac{25}{32}[/tex]

Dividing both the terms, we get,

[tex]\frac{a_{4}}{a_{9}}=\frac{-25}{\frac{25}{32}}[/tex]

i.e. [tex]\frac{ar^{3}}{ar^{8}}=\frac{-25}{\frac{25}{32}}[/tex]

i.e. [tex]\frac{1}{r^{5}}=-32[/tex]

i.e. [tex]r^5=\frac{-1}{32}[/tex]

i.e. [tex]r^5=-0.03125[/tex]

i.e. r= -0.5

So, we get,

[tex]a_{4}=ar^{3}=-25[/tex]

i.e. [tex]a(-0.5)^{3}=-25[/tex]

i.e. [tex]a=\frac{-25}{-0.125}[/tex]

i.e. a= 200.

Then, we have,

[tex]a_{15}=ar^{15-1}[/tex]

i.e. [tex]a_{15}=ar^{14}[/tex]

i.e. [tex]a_{15}=200\times (-0.5)^{14}[/tex]

i.e. [tex]a_{15}=200\times 6.1\times 0.00001[/tex]

i.e. [tex]a_{15}=1220\times 0.00001[/tex]

i.e. [tex]a_{15}=0.122[/tex]

Thus, the 15th term of the geometric sequence is 0.0122

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