1.

A sequence is defined as follows : `a_(1)=3, a_(n)=2a_(n-1)+1`, where `n gt 1`. Where `n gt 1`. Find `(a_(n+1))/(a_(n))` for n = 1, 2, 3.

Answer» `a_(1)=3`
`a_(n)=2a_(n-1)+1` where `n gt 1`
put n=2, we get
`a_(2)=2a_(1)+1=2xx3+1=7`
put n=3, we get
`a_(3)=2a_(2)+1=2xx7+1=15`
put n=4, we get
`a_(4)=2a_(3)+1=2xx15+1=31`
Now, for n=1
`(a_(n+1))/(a_(n))=(a_(2))/(a_(1))=(7)/(3)`
For n=2,
`(a_(n+1))/(a_(n))=(a_(3))/(a_(2))=(15)/(7)`
For n=3,
`(a_(n+1))/(a_(n))=(a_(4))/(a_(3))=(31)/(15)`


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