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Published in category Mathematics, 16.10.2020 >> <<

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

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\bf \begin{array}{llccll}&#10;term&value\\&#10;\text{\textemdash\textemdash\textemdash}&\text{\textemdash\textemdash\textemdash}\\&#10;a_1&-4096\\&#10;a_2&-4096r\\&#10;a_3&-4096rr\\&#10;a_4&-4096rrr\\&#10;&-4096r^3\\&#10;&64&#10;\end{array}\implies -4096r^3=64&#10;\\\\\\&#10;r^3=\cfrac{64}{-4096}\implies r^3=-\cfrac{1}{64}\implies r=\sqrt[3]{-\cfrac{1}{64}}&#10;\\\\\\&#10;r=\cfrac{\sqrt[3]{-1}}{\sqrt[3]{64}}\implies \boxed{r=\cfrac{-1}{4}}\\\\&#10;-------------------------------

\bf n^{th}\textit{ term of a geometric sequence}\\\\&#10;a_n=a_1\cdot r^{n-1}\qquad &#10;\begin{cases}&#10;n=n^{th}\ term\\&#10;a_1=\textit{first term's value}\\&#10;r=\textit{common ratio}\\&#10;----------\\&#10;r=-\frac{1}{4}\\&#10;a_1=-4096\\&#10;n=6&#10;\end{cases}&#10;\\\\\\&#10;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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Answer by Guest


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.


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