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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.

User Balexandre
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1 Answer

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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
User Ellet
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