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Find a formula for 1 2 + 1 4 + 1 8 +···+ 1 2n by examining the values of this expression for small values of n.
b.prove the formula you conjectured in part (a)

Respuesta :

LammettHash
[tex]S_n=\dfrac12+\dfrac14+\dfrac18+\cdots+\dfrac1{2^n}[/tex]
[tex]\dfrac12S_n=\dfrac1{2^2}+\dfrac1{2^3}+\dfrac1{2^4}+\cdots+\dfrac1{2^{n+1}}[/tex]
[tex]\implies S_n-\dfrac12S_n=\dfrac12-\dfrac1{2^{n+1}}[/tex]
[tex]\implies S_n=1-\dfrac1{2^n}[/tex]

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