A super-Fibonacci sequence is a list of whole numbers with the property that, from the third term onwards, every term is the sum of all the previous terms. For example,

1, 4, 5, 10, ...
How many super-Fibonacci sequences with 1 involve the number 2016?

A.1
B.3
C.5
D.7
E.9

If I read this correctly, we would get:

1, 4, 5, 10, 20, 40, 80, ...
in effect, each term is twice its previous term from the 4th term on

Where does the number 2016 come in?
clearly not in this sequence.

suppose my term start with 1, 2 , then we get
1,2, 3, 6, 12, 24, , again we double each time
we will hit 2016

starting with 1, 3, we get
1,3,4,8,16, ... same doubling pattern

notice in all cases, the terms after the third are multiples of the third term.
So the third term establishes the multiples
2016 = 63(2^5)
= (9)(7)(2^5)

so as long as the third terms starts with
63, 63(2), 63(4), 63(8), 63(16) or 63(32)
or
63, 126, 252, 504, 1008, 2016 it will work

1, 62, 63, 126, 252, 504, 1008, 2016, ... YUP
or
1, 125, 126, 252, ..., 2016
1, 251, 252, ... , 2016
...
1, 2015, 2016

I count 6 of those.

But what about 1, 2016, 2017, .... ?
wouldn't that also involve 2016?

So I would go with 7

Very interesting question.

Thanks so much for giving a great explanation

2i dimensional analysis is what I'm trying to do, please tell me im wrong

Hmm, let me put on my mathematician clown hat for a moment. So we're looking for super-Fibonacci sequences that involve the number 2016, right? Well, to solve this problem, we need to keep adding up the previous terms to get the next term. So let's see if we can come up with some sequences!

Starting with 1, the first term is 1. Since we can't add anything to get to 2016, this sequence doesn't work. So we move on.

Next, we try starting with 4. The second term must be 1 (because it's greater than all the previous terms), and the third term is 5 (the sum of 4 and 1). From there, we add 4 and 5 to get 9, then 4 and 5 and 9 to get 18, and so on. But does this sequence involve 2016? Let me do some clown math here...

4 + 1 + 5 + 9 + 18 + 37 + 75 + 152 + 307 + 618 + 1239 + 2470 = 4965

Oh clown noodles! We went way over 2016. Alright, let's try another one.

How about starting with 5? The second term must be 1, and the third term is 6. And we keep adding the previous terms to get the next one. Let's see if this one takes us to 2016...

5 + 1 + 6 + 12 + 24 + 49 + 99 + 198 + 397 + 796 + 1595 = 4981

Hey, 2016 is still clowning around somewhere. We're not quite there yet.

Let's try one more, just for fun. How about starting with 10? The second term must be 1, and the third term is 11. And we keep adding the previous terms to get to the next one.

10 + 1 + 11 + 22 + 44 + 89 + 179 + 359 + 719 + 1439 + 2879 = 5732

Well, 2016 is definitely not showing up at this circus either.

After checking a few more sequences, it seems like none of them involve the number 2016. So, my dear mathematician, the answer is A.1. There is only one super-Fibonacci sequence that involves the number 2016.

To determine how many super-Fibonacci sequences with 1 involve the number 2016, we first need to understand how the super-Fibonacci sequence works.

A super-Fibonacci sequence is defined as a list of whole numbers where every term from the third term onwards is the sum of all the previous terms.

Let's try to generate the super-Fibonacci sequence starting with 1:

1, (1)
1, (1, 1)
1, (1, 1, 2)
1, (1, 1, 2, 4)
1, (1, 1, 2, 4, 8)
...

In the sequence above, the terms inside parentheses represent the previous terms being summed to produce the current term.

From the given question, we need to find out how many super-Fibonacci sequences involving the number 2016 exist, considering 1 as the first term.

To do this, we can generate super-Fibonacci sequences, adding up terms from the previous sequences until we reach or exceed 2016.

Let's proceed with generating the super-Fibonacci sequence:

1, (1)
1, (1, 1)
1, (1, 1, 2)
1, (1, 1, 2, 4)
1, (1, 1, 2, 4, 8)
1, (1, 1, 2, 4, 8, 16)
1, (1, 1, 2, 4, 8, 16, 32)
1, (1, 1, 2, 4, 8, 16, 32, 64)
1, (1, 1, 2, 4, 8, 16, 32, 64, 128)
1, (1, 1, 2, 4, 8, 16, 32, 64, 128, 256)
1, (1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512)
1, (1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024)
1, (1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048)

Looking at the generated sequence, we can see that 2016 is not present in any of the super-Fibonacci sequences.

Therefore, the answer to the question is: 0 super-Fibonacci sequences involve the number 2016 when considering 1 as the first term.

The correct option is: A. 1

1 or 0, 1 part = i

2016
1016
06
03
01
0(1)
Modulus division by prime numbers baby