What is the seventh term of the geometric sequence where a1=-4,096 a4=64 ?

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Feb 9, 2018

Answer:

The seventh term of the geometric sequence is
#a7=-1#

Explanation:

Given:
#a1=-4096#
#a4=64#
To find:
#a7=?#

We know that,

In geometric progression, the successive terms are in constant ratio.

Let r be the constant ratio.

1st term #a1=-4096#
2nd term #a2=-4096r#
3rd term #a3=-4096r^2#
4th term #a4=-4096r^3#

Equating #a4=64#

Simplifying,
#-4096r^3=64#
#r^3=64/-4096#
#64=4^3#
#-4096=(-16)^3#

Thus,
#r^3=4^3/(-16)^3#

By the law of indices
#4^3/(-16)^3=(4/(-16))^3#

Now,
#r^3=(4/(-16))^3#

Simplifying
#r=4/-16=1/-4=-1/4#

The common ratio is thus, #-1/4#
5th term #a5=64(-1/4)=-16#
6th term#a6=-16(-1/4)=4#
7th term #a7=4(-1/4)=-1#

Listing the series
#a1=-4096#
#a2=-4096(-1/4)=1024#
#a3=1024(-1/4)=-256#
#a4=-256(-1/4)=64#, checked
#a5=64(-1/4)=-16#
#a6=-16(-1/4)=4#
#a7=4(-1/4)=-1#

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