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# Advanced Math/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 formula you describe is just a way of calculating the next value in the sequence (at index n+1) based on the previously calculated value (at n). So if you know the value at a particular index (n=0 in your case), you can immediately calculate the next value (incrementing n by one, i.e., n -> n+1). You can then calculate the value at n+2 and then, once you've got the value at n+2, the value at n+3 and so on.

Let me write the quantity "{ a subscript n }" as A[n], so that your formula can be written as A[n+1] = 2A[n] + 2. For your case, for n=0 you have A[0] = 1. Plugging this in the formula gives

A[1] = 2A[0]+2 = 2(1)+2 = 4.  <-  note the difference between the [ ] and the ( ) brackets!; [ ] means subscript, ( ) means quantity to multiply by

Incrementing n gives n+1 -> 0+1 = 1. Plugging in the value you just calculated for n=1, namely A1=4, gives

A[2] = 2A[1]+2 = 2(4)+2 = 10.

Doing this again gives

A[3] = 2A[2]+2 = 2(10)+2 = 20+2 = 22.

So there you have it for A[3].

The formula for the sequence is called a recursion formula. In general, there's no reason that you couldn't have more terms in the recursion formula, say

A[n+1] = aA[n] + bA[n-1] + ... + qA[n-k] + C, where a,b,...,q and C are constants, and k=integer.

Hope this helps.
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