Question

Geometric or arithmetic or neither ?

Answer 1

Sequence 1 is an 'Arithmetic but not Geometric Sequence'

Sequence 2 is 'Geometric but not Arithmetic Sequence'

Explanation:

We know that,

1. Arithmetic Sequence is a sequence in which the difference of one term and the next term is a same constant for all terms.

2. Geometric Sequence is a sequence in which the division of two terms gives the same value for all terms.

Now, we check the above properties in the given options,

In Sequence 1 i.e.
`(1)/(2) , (7)/(6) ,(11)/(6) ,(5)/(2)` , . . . .

We see that the difference between the terms comes out to be
`(2)/(3)`,

for eg.
`(7)/(6) - (1)/(2)` =
`(4)/(6) = (2)/(3)`

But, the division of two terms gives different values,

for eg.
`((7)/(6) )/((1)/(2) ) = (7)/(3)` and
`((11)/(6) )/((7)/(6) ) = (11)/(7)`

Hence, this sequence is not a Geometric Sequence but an Arithmetic Sequence.

In Sequence 2 i.e.
`(1)/(2) , (1)/(3) ,(2)/(9) ,(4)/(27)` , . . . .

We see that the difference of terms is not same constant but are different values,

for eg.
`(1)/(3) - (1)/(2)` =
`(-1)/(6)` and
`(1)/(3) - (2)/(9)` =
`(1)/(9)`

But, the division of different terms gives same constant i.e.
`(2)/(3)`,

for eg.
`((1)/(3) )/((1)/(2) ) = (2)/(3)`.

Hence, this sequence is not a Arithmetic Sequence but a Geometric Sequence.