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lesya692 [45]
2 years ago
12

What is the 6th term of the geometric sequence where a1 = -4096 and a4 = 64?

Mathematics
2 answers:
photoshop1234 [79]2 years ago
5 0

Answer:

The sixth term of the geometric sequence is 4.

Step-by-step explanation:

Given : Geometric sequence term are a_1=-4096 and  a_4=64    

To find : What is the 6th term of geometric sequence ?

Solution :

a_1=-4096

We know, fourth term is a_4=ar^3

a_4=64    

So, 64=(-4096)r^3

\frac{64}{(-4096)}=r^3

-0.015625=r^3

r=(-0.015625)^{\frac{1}{3}}

The sixth term is  a_6=ar^5

Substitute,

a_6=(-4096)((-0.015625)^{\frac{1}{3}})^5

a_6=-4096((-0.015625)^{\frac{5}{3}})

a_6=-4096(-0.0009765625)

a_6=4

Therefore, The sixth term of the geometric sequence is 4.

Akimi4 [234]2 years ago
4 0
\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
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