Recurrence formula

Question

Could you please help me to understand the task.

The topic of the task is Recurrence formula. There is a txt file with integers that represent the members of the row of a(k), starting with k = 0. It is required to calculate the value of S (100), starting with k = 3, if S (k) = S (k-1) + 2 * a (k + 1) + a (k) and S (2) = 5.

As far as I understand the conditions of the task, each namber is a(k), where k = 0, ..., n. If this is correct, then the first number is a(0) = 197, the second number is a(1) = 381, the third is a(2) = 171, etc. I don't understand what S(k) is.

By definition, a recurrence formula is a formula that expresses each term in a sequence through the previous terms. Does this mean that S (100) is the hundredth term in the sequence of numbers? If so, then why was it expressed through the new letter S(k), and not left a(k)? It turns out that the formula contains both S(k) and a(k), denoting the same thing, which seems to be nonsense. And again, if S(k) is the k-th term, then why S(2) = 5? Where does 5 come from?

For some reason, the name of the text file with numbers contains the word sum - sum_recurr_rd_1.txt. Does this mean that S(k) is not the k-th term, but the sum of the k-th terms and I need to find the sum of the first 100 numbers but starting from a(3) = 385 instead of a(0) = 197 since it is said to start with k = 3. But even if we assume that this is the case (although I'm not sure), then the result of summ of firts 100 terms is not the correct answer. I checked. Or maybe I need to sort the sequence of numbers before starting the calculation? Although nothing is said about this.

I see that my understanding of the assignment and perhaps the topic itself is not correct. Please help me figure out what is the S(k) that need to be find?

Thank you!

score 0 · Accepted Answer · answered Mar 21 '21 at 18:51

In the context, it is reasonable to assume S(k) is the sum of first k terms and a(k) is each term. We need to know S(2) (or a(1)+a(2)),a(3),a(4) and a(101) to get S(100)as proved below. Not sure how to link this with the data, as you said it is not clear where S(2)=5 comes from.

S(k)=S(k-1)+2a(k+1)+a(k)

S(100)=S(99)+2a(101)+a(100)

S(99)=S(98)+2a(100)+a(99)

S(98)=S(97)+2a(99)+a(98)

...

S(4)=S(3)+2a(5)+a(4)

S(3)=S(2)+2a(4)+a(3)

So

S(100)=S(2)+2a(101)+2a(4)+a(3)+3(a(4)+a(5)+...+a(100))

S(100)=S(2)+2a(101)+2a(4)+a(3)+3S(100)-3(a(1)+a(2)+a(3))

S(100)=-$\frac{S(2)+2a(101)+2a(4)+a(3)-3(a(1)+a(2)+a(3))}{2}$

S(100)=S(2)-a(101)-a(4)+a(3)

S(100)=S(2)+a(3)-a(4)-a(101)

thank you! I got it. This formula is for finding terms in a new set of numbers. I thought it was for the existing set of numbers that are given in the problem. — Gelios, Mar 22 '21 at 07:54

Recurrence formula

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