Question

Can someone give me the answers and step by step instructions please??

Answer 1

`-1,4,-7,10,...` neither

`192,24,3,(3)/(8),...` geometric progression

`-25,-18,-11,-4,...` arithmetic progression

Explanation:

Given:

sequences:
`-1,4,-7,10,...`

`192,24,3,(3)/(8),...`

`-25,-18,-11,-4,...`

To find: which of the given sequence forms arithmetic progression, geometric progression or neither of them

Solution:

A sequence forms an arithmetic progression if difference between terms remain same.

A sequence forms a geometric progression if ratio of the consecutive terms is same.

For
`-1,4,-7,10,...`:

`4-(-1)=5\\-7-4=-11\\10-(-7)=17\\So,\,\,4-(-1)\\eq -7-4\\eq 10-(-7)`

Hence,the given sequence does not form an arithmetic progression.

`(4)/(-1)=-4\\(-7)/(4)=(-7)/(4)\\(10)/(-7)=(-10)/(7)\\So,\,\,(4)/(-1)\\eq (-7)/(4)\\eq (10)/(-7)`

Hence,the given sequence does not form a geometric progression.

So,
`-1,4,-7,10,...` is neither an arithmetic progression nor a geometric progression.

For
`192,24,3,(3)/(8),...` :

`(24)/(192)=(1)/(8)\\(3)/(24)=(1)/(8)\\((3)/(8))/(3)=(1)/(8)\\So,\,\,(24)/(192)=(3)/(24)=((3)/(8))/(3)`

As ratio of the consecutive terms is same, the sequence forms a geometric progression.

For
`-25,-18,-11,-4,...` :

`-18-(-25)=-18+25=7\\-11-(-18)=-11+18=7\\-4-(-11)=-4+11=7\\So,\,\,-18-(-25)=-11-(-18)=-4-(-11)`

As the difference between the consecutive terms is the same, the sequence forms an arithmetic progression.