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

Given:
a₁ = 1
$a_{n} = \frac{1}{2} a_{n-1},\,n\ \textgreater \ 1$
Therefore
a₂ = 1/2
a₃ = (1/2)*(1/2) = (1/2)²
...
$a_{n} = (1/2)^{n-1}$

Also,
b₁ = a₁
$b_{n} = b_{n-1}+a_{n}$
Therefore
b₁ = 1
b₂ = b₁ + a₂ = 1 + (1/2)
b₃ = b₂ + a₃ = 1 + 1/2 + (1/2)²
...
$b_{n} = 1 +\Sigma_{k=1}^{n-1} \,(1/2)^{k}$
This is the sum of a geometric sequence with common ratio r=1/2.

The 50th term is
$b_{50} = 1 + \frac{(1/2)[1-(1/2)^{49}]}{1-(1/2)} =2$

The 1000000th term is
$b_{1000000} = 1 + \frac{(1/2)[1-(1/2)^{999999}]}{1-(1/2)}=2$