Question

Suppose that we have the following sequence :

a_1=1
 a_n=1/2 a_(n-1) , for n>1
 
We define a new sequence as such :
b_1=a_1
b_n=b_(n-1)+a_n
     
Find the 50th term of the sequence b_n.
Find the 1000000th term of the sequence b_n.

Answer 1

We have

`a_(1)=1` which is the first term in the sequence

`a_(n) = (1)/(2) a_(n-1)` where
`n` is the term in the sequence


`b_(1) = a_(1) = 1` which is the first term in the
`b_(n)` sequence

`b_(n)= b_(n-1)+ a_(n)`


`50^(th)` term of
`b_(n)`

`b_(50) = b_((50-1))+a_(50)`

`b_(50) = b_(49)+ a_(50)`

`b_(50) = b_(49)+( (1)/(2) a_(49))`


`b_(49) = 49` - we work this out from the first term
`b_(1) =1`

`a_(49)=49` - we work this out from the first term
`a_(1)=1`

`a_(50) = (1)/(2)(49)`

Hence
`b_(50)=49+( (1)/(2))49 = 73.5`

The
`1000000^(th)` term is

`b_(1000000) = b_((1000000-1)) +a_(1000000)`

`b_(1000000)= b_(999999) + a_(1000000)`


`b_(999999) = 999999` from
`b_(1) =1`

`a_(1000000) = (1)/(2)( a_(999999) = (1)/(2) (999999)`

Hence,

`b_(1000000)=999999+ ( (1)/(2))999999 = 1499998.5`