Sequences!!!?
In the non-decreasing sequence of odd integers {A(1), A(2), A3,...}={1,3,3,3,5,5,5,5,5,...} each odd positive integer k appears k times. It is a fact that there are integers b,c, and d such that for all positive integers n, A(n)=b[(n+c)^(1/2)] +d, where [x] denotes the largest integer not exceeding x. what does the sum b+c+d equal?
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Note that A(1)=1, A(4)=3, A(9)=5,..., A(n²)=2n-1,....
That tells you that A(n) is something like 2√n - 1, but that formula is exact only for square n.
Now let's experiment to see what fixing up we need to do to get it exactly right.
Here's the sequence 2√n - 1 starting at n=1:
1, 1.83, 2.46, 3, 3.47, ...
So we need to boost the value for nonsquare n's.
That would argue for using a positive c,
but that is doomed to fail:
we'll need larger and larger values of c as n gets bigger.
Instead, we'll take c=-1 and then add back what we need in the form of d.
Try A(n) = 2[ √(n-1) ] + 1. Oh, yeah!
So b+c+d=2 + (-1) + 1 = 2.