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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.

Respuesta :

LammettHash
[tex]a_n=\dfrac12a_{n-1}[/tex]
[tex]a_n=\dfrac1{2^2}a_{n-2}[/tex]
[tex]a_n=\dfrac1{2^3}a_{n-3}[/tex]
[tex]a_n=\cdots=\dfrac1{2^{n-1}}a_1[/tex]
[tex]a_n=\dfrac1{2^{n-1}}[/tex]

[tex]b_n=b_{n-1}+\dfrac1{2^{n-1}}[/tex]
[tex]b_n=b_{n-2}+\dfrac1{2^{n-1}}+\dfrac1{2^{n-2}}[/tex]
[tex]b_n=b_{n-3}+\dfrac1{2^{n-1}}+\dfrac1{2^{n-2}}+\dfrac1{2^{n-3}}[/tex]
[tex]b_n=\cdots=b_1+\dfrac1{2^{n-1}}+\dfrac1{2^{n-2}}+\cdots+\dfrac12[/tex]
[tex]b_n=a_1+\displaystyle\sum_{k=1}^{n-1}\frac1{2^{n-k}}[/tex]
[tex]b_n=1+\displaystyle\sum_{k=1}^{n-1}\frac1{2^{n-k}}[/tex]
[tex]b_n=\displaystyle\sum_{k=1}^n\frac1{2^{n-k}}[/tex]
[tex]b_n=\displaystyle\frac1{2^n}\underbrace{\sum_{k=1}^n2^k}_{S_n}[/tex]

[tex]S_n=1+2+2^2+\cdots+2^{n-1}+2^n[/tex]
[tex]\implies2S_n=2+2^2+2^3+\cdots+2^n+2^{n+1}[/tex]
[tex]\implies S_n-2S_n=-S_n=1-2^{n+1}[/tex]
[tex]\implies S_n=2^{n+1}-1[/tex]

[tex]b_n=\dfrac{2^{n+1}-1}{2^n}=2-\dfrac1{2^n}[/tex]

[tex]\implies b_{50}=2-\dfrac1{2^{50}}\approx1.99999999999999911182158[/tex]

[tex]\implies b_{10^6}=2-\dfrac1{2^{10^6}}\approx2.00000000000000000000000[/tex]

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