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# Compute the limit of the recursive sequence an+1 = (an + an-1) / 2

Let be the recursively defined sequence

where the first two terms are given and .

1. Assume that this sequence converges and compute its limit in terms of the initial terms and .
2. Prove that the sequences converges for every choice of and . Assume .

1. Assume the limit exists, say

Then, we have

Hence, the value does not depend on , and we have

Now, taking the limit

2. Proof. Assume and let . Now, we claim for that

We prove this claim by induction. For the case we have

So the statement is indeed true for the case . Now, assume the statement is true for some integer . Then we have

Therefore, the proposed formula indeed holds for all integers . We then express this formula as a sum,

So the terms of the sequence are the partial sums of this series. But, since is a constant (for any given and ) this is a geometric series; hence, it converges. Therefore, the terms of the sequence converge to a finite limit