We have
[tex] a_{1}=1 [/tex] which is the first term in the sequence
[tex] a_{n} = \frac{1}{2} a_{n-1} [/tex] where [tex]n[/tex] is the term in the sequence
[tex] b_{1} = a_{1} = 1[/tex] which is the first term in the [tex] b_{n} [/tex] sequence
[tex] b_{n}= b_{n-1}+ a_{n} [/tex]
[tex]50^{th} [/tex] term of [tex]b_{n} [/tex]
[tex]b_{50} = b_{(50-1)}+a_{50} [/tex]
[tex]b_{50} = b_{49}+ a_{50} [/tex]
[tex] b_{50} = b_{49}+( \frac{1}{2} a_{49}) [/tex]
[tex]b_{49} = 49 [/tex] - we work this out from the first term [tex]b_{1} =1[/tex]
[tex]a_{49}=49 [/tex] - we work this out from the first term [tex]a_{1}=1 [/tex]
[tex]a_{50} = \frac{1}{2}(49) [/tex]
Hence [tex]b_{50}=49+( \frac{1}{2})49 = 73.5 [/tex]
The [tex]1000000^{th} [/tex] term is
[tex]b_{1000000} = b_{(1000000-1)} +a_{1000000} [/tex]
[tex]b_{1000000}= b_{999999} + a_{1000000} [/tex]
[tex]b_{999999} = 999999[/tex] from [tex]b_{1} =1[/tex]
[tex] a_{1000000} = \frac{1}{2}( a_{999999} = \frac{1}{2} (999999)[/tex]
Hence,
[tex]b_{1000000}=999999+ ( \frac{1}{2})999999 = 1499998.5 [/tex]
