Mathematics, 08.11.2019 04:31 alinegonzalez0027

What is the 6th term of the geometric sequence where a1 = −4,096 and a4 = 64?

Answers: 2

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Answers

First, we need to solve for the common ratio from the data given by using the equation.

a(n) = a(1) r^(n-1)
64 = -4096r^(4-1)
64 = -4096r^3
r = -1/4

Then, we can find the 6th term as follows:

an = -4096 r^(6-1)
an = -4096(-1/4)^5

an = 4

Answer from: oreoassassin38

$a_1=4.096;\ a_4=64\\\\a_4:a_1=r^3\\\\r^3=64:4.096\\\\r^3=15.625\\\\r=\sqrt[3]{15.625}\\\\r=2.5\\\\a_6=a_4r^2\to a_6=64\cdot2.5^2=64\cdot6.25=400\\\\\boxed{a_6=400}$

Answer from: yejinschoi6362

$\bf \begin{array}{llccll}
term&value\\
\text{\textemdash\textemdash\textemdash}&\text{\textemdash\textemdash\textemdash}\\
a_1&-4096\\
a_2&-4096r\\
a_3&-4096rr\\
a_4&-4096rrr\\
&-4096r^3\\
&64
\end{array}\implies -4096r^3=64
\\\\\\
r^3=\cfrac{64}{-4096}\implies r^3=-\cfrac{1}{64}\implies r=\sqrt[3]{-\cfrac{1}{64}}
\\\\\\
r=\cfrac{\sqrt[3]{-1}}{\sqrt[3]{64}}\implies \boxed{r=\cfrac{-1}{4}}\\\\
-------------------------------$

$\bf n^{th}\textit{ term of a geometric sequence}\\\\
a_n=a_1\cdot r^{n-1}\qquad 
\begin{cases}
n=n^{th}\ term\\
a_1=\textit{first term's value}\\
r=\textit{common ratio}\\
----------\\
r=-\frac{1}{4}\\
a_1=-4096\\
n=6
\end{cases}
\\\\\\
a_6=-4096\left( -\frac{1}{4} \right)^{6-1}\implies a_6=-4^6\left( -\frac{1}{4} \right)^5$

Answer from: tutyfruity29

The 6th term of the geometric sequence is, $0.125=\frac{1}{8}$

Step-by-step explanation:

The nth term of the geometric sequence is given by:

$a_n = a_1 \cdot r^{n-1}$

where,

a_1 is the first term

r is the common ratio of the term

n is the number of terms

As per the statement:

a_1 = 128 and a_3 = 8

For n = 3

$a_3 = a_1 \cdot r^2$

Substitute the given values we have;

$8 = 128 \cdot r^2$

Divide both sides by 128 we have;

$\frac{1}{16} = r^2$

⇒ $\sqrt{\frac{1}{16}} =r$

⇒ $\frac{1}{4} = r$

$r =\frac{1}{4}=0.25$

We have to find the 6th term of the geometric sequence.

For n = 6 we have;

$a_6 = a_1 \cdot r^5$

⇒ $a_6 = 128 \cdot (0.25)^5 = 128 \cdot 0.0009765625$

Simplify:

$a_6 =0.125=\frac{1}{8}$

Therefore, the 6th term of the geometric sequence is, $0.125=\frac{1}{8}$

Answer from: lzjsjdhdgd442

−57,395,628

Step-by-step explanation:

The n-th term of a geometric sequence is given by ...

an = a1·r^(n-1)

Then the ratio of the 8th term to the 1st term is ...

a8/a1 = (a1·r^(8-1))/(a1·r^(1-1)) = r^7

and the common ratio is ...

(-8748/4)^(1/7) = -3

That makes the 16th term be ...

a16 = 4·(-3)^(16 -1) = -57,395,628

Answer from: aamu15

0.2

hope this helps

Answer from: zoewilliamss26

6th term is 0.125

Step-by-step explanation:

a1 = 128 and a3 = 8

To find nth term of geometric sequence we use formula

$a_n = a_1(r)^{n-1}$

a1 is the first term

r is the common ratio

a1= 128

Given a3 = 8, n=3, a1= 128

$a_n = a_1(r)^{n-1}$

$a_3 = 128(r)^{3-1}$

$8 = 128(r)^{2}$

Divide by 128 on both sides

1/16 = r^2

take square root on both sides

So r= 1/4

To find 6th term, replace r= 1/4

$a_6 = 128(\frac{1}{4})^{6-1}$

a_6= 0.125

Answer from: preciadogabriel40

Option 3rd is correct

a_6 = -1

Step-by-step explanation:

The nth term for the geometric sequence is given by:

$a_n = a_1 \cdot r^{n-1}$

where,

a_1 is the first term

r is the common ratio

n is the number of terms.

As per the statement:

a_1 = 1024

a_4 = -16

For n = 4, we have;

$a_4 = a_1 \cdot r^3$

⇒ $-16 = 1024 \cdot r^3$

Divide both sides by 1024 we have;

$-\frac{1}{64} =r^3$

⇒ $r =\sqrt[3]{-\frac{1}{64}}=\sqrt[3]{-\frac{1}{4^3}}$

⇒ $r = -\frac{1}{4} = -0.25$

We have to find the value of 6th term.

for n = 6

$a_6 = 1024 \cdot (-0.25)^5 = 1024 \cdot (-0.0009765625) = -1$

⇒ a_6 = -1

Therefore, the 6th term of the geometric sequence is, -1

Answer from: shermoisllo3

r = -3

16th term: -516560652

Step-by-step explanation:

Let's use the formula for a geometric sequence:

, where is the nth term, is the first term, and r is the common ratio

Here, we know the first term is 4. To find r, let's plug in 8 for n and -8748 for :

-8748 = 4 *

= -2187

r = -3

Answer from: auraziabagley

r = -3

16th term: -516560652

Step-by-step explanation:

Let's use the formula for a geometric sequence:

$a_n=a_1r^{n-1}$ , where a_n is the nth term, a_1 is the first term, and r is the common ratio

Here, we know the first term is 4. To find r, let's plug in 8 for n and -8748 for a_n :

$a_n=a_1r^{n-1}$

-8748 = 4 * $r^{8-1}$

-8748 = 4 * r^7

r^7 = -2187

r = -3

So we know that the common ratio is -3. Now, we want to find the 16th term, so n = 16:

$a_n=a_1r^{n-1}$

$a_{16}=a_1*(-3)^{16-1}=4*(-3)^{17}=-516560652$

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