Question

Geometric or arithmetic or neither ?

Answer 1

Answer:

Sequence 1 is an 'Arithmetic but not Geometric Sequence'

Sequence 2 is 'Geometric but not Arithmetic Sequence'

Step-by-step 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. $\frac{1}{2} , \frac{7}{6} ,\frac{11}{6} ,\frac{5}{2}$ , . . . .

We see that the difference between the terms comes out to be $\frac{2}{3}$,

for eg. $\frac{7}{6} - \frac{1}{2}$ = $\frac{4}{6} = \frac{2}{3}$

But, the division of two terms gives different values,

for eg. $\frac{\frac{7}{6} }{\frac{1}{2} } = \frac{7}{3}$ and $\frac{\frac{11}{6} }{\frac{7}{6} } = \frac{11}{7}$

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

In Sequence 2 i.e. $\frac{1}{2} , \frac{1}{3} ,\frac{2}{9} ,\frac{4}{27}$ , . . . .

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

for eg. $\frac{1}{3} - \frac{1}{2}$ =  $\frac{-1}{6}$ and $\frac{1}{3} - \frac{2}{9}$ = $\frac{1}{9}$

But, the division of different terms gives same constant i.e. $\frac{2}{3}$,

for eg. $\frac{\frac{1}{3} }{\frac{1}{2} } = \frac{2}{3}$.

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