Question

How do you write an explicit formula for this sequence: 2, -1, 1/2, -1/4, 1/8?

Answer 1

This sequence is in GP
Since
The ratio of `"2nd Term"/"1st Term""=-1/2`

The ratio of `"3rdTerm"/"2nd Term"=(1/2)/-1=-1/2`

The ratio of `"4thTerm"/"3rdTerm"=(-1/4)/(1/2)=-1/2`
and so on
So the sequence has 1st term 2
and common ratio `-1/2`

If 1st term of GP be A and common ratio is r then its nth term is given by `t_n=ar^(n-1)`

If we put a =2 , r = `-1/2 and n=1` we get `t_1=2xx(-1/2)^(1-1)=2`

If we put a =2 , r = `-1/2 and n=2` we get `t_2=2xx(-1/2)^(2-1)=-1`

If we put a =2 , r = `-1/2 and n=3` we get `t_3=2xx(-1/2)^(3-1)=1/2`