Let a n + 1 = 2 a n + 1 a n + 1 .For a 1 =1 can we show that a n &lt;2

Question

Let       a          n      +      1        =            2              a                  n                    +      1                      a                  n                    +      1       .For       a          1      =1 can we show that       a          n      &amp;lt;2

emarisidie6 · Accepted Answer

a          n      +      1        =            2              a        n            +      2      −      1                      a        n            +      1        =  2  −      1                  a        n            +      1        &amp;lt;  2.Also,       a    1    &amp;lt;  2

vagnhestagn · Accepted Answer

1  &amp;lt;  2,  2      a    n    +  1  &amp;lt;  2      a    n    +  2 as long as       a    n   are positive, so   (  2      a    n    +  1  )      /    (      a    n    +  1  )  &amp;lt;  2 and you are done.

Let a_(n+1)=(2a_(n)+1)/(a_(n)+1).For a1=1 can we show that a_n < 2

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