Question

Question #daf71

Answer 1

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`