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Algebra/sequences and series

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Question
The sequence { a subscript n } is defined by ( a subscript 0 ) = 1
and ( a subscript (n+1) ) = (2a subscript n) + 2 for  n = 0,1,2,3,... .

What is the value of ( a subscript 3 ) ??

I've been having trouble understanding this. Could you explain this to me ?  Thanks.

Answer
The series is actually explained by a recurrence relation. In other words, if you know what ( a subscript 0 ) is, you can discover ( a subscript 1 ). If you know what ( a subscript 1 ) is, you can discover ( a subscript 2 ), so on and so forth.



By the Recurrence relation given in the second line of your problem,

we can say for n=0,  ( a subscript 1 ) =2*( a subscript 0 )  +2

Substituting in ( a subscript 0 ) =1

gives ( a subscript 1 ) = 2 * 1 +2 = 3



Now, for n=1, let us extend this recurrent relation such that

( a subscript 2 ) =2*( a subscript 1 ) +2 = 2* 3 + 1 = 7    



Doing this once again for n=2:

( a subscript 3 ) =2*( a subscript 2 )  +2 = 2* 7 +1 = 15 (shown)

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