Question

Write the rule for the nth term of the following sequence.

1 over 4 comma 1 over 2 comma 1 comma 2 comma 4 comma ...

Answer 1

`2^(n-3)` is the nth term of the given sequence.

Explanation:

The sequence given is

`(1)/(4),(1)/(2),1,2,4 .....`

We can clearly see that 1st term is
`(1)/(4)` and 2nd term is
`(1)/(2)`

2nd term is obtained by multiplying the 1st term by 2.

2nd term is
`(1)/(2)` and 3rd term is 1.

3rd term is obtained by multiplying the 2nd term by 2.

3rd term is 1 and 4th term is 2.

4th term is obtained by multiplying the 3rd term by 2.

Clearly, the given series is a Geometric Progression(GP) with

First term,
`a = (1)/(4)`

Common Ratio,
`r = 2`

We know that
`n^(th)` term for a GP is:

`a_n = ar^(n-1)`

Putting values of a and r

`a_n = (1)/(4)2^(n-1)\\a_n = (1)/(2^(2) )2^(n-1)\\a_n = 2^(n-1-2)\\a_n = 2^(n-3)`

Hence,
`2^(n-3)` is the nth term of the given sequence.