
David W. answered 04/20/17
Experienced Prof
This is a Recursive Definition of a sequence of numbers:
T1 = 1 \
T2 = 2 / base case
Tn = Tn-1 + Tn-2 -- recursive case
This particular sequence is called the Fibonacci Sequence.
Let’s expand T4 as an example of this recursion:
T4 = T3 + T2 [use recursive case on T4]
T4 = ( T2 + T1 ) + T2 [use recursive case on T3]
T4 = ( 2 + T1 ) + T2 [use base case for T2]
T4 = 3 + 2 [so, T3 = 3, right?]
T4 = 5
It gets longer, but not more complicated (especially for a very fast computer), when values have to be expanded recursively (that is, over and over), but there are many, many useful applications of recursion instead of iteration (simple looping).
Now, what about the remainder of ( T21 / 6 ) ???
Using just the numbers in the series:
1 2 3 5 8 13 21 34 55 89
144 233 377 610 987 1597 2584 4181 6765 10946
The remainder of 17711/6 is 5.
Observing the remainders:
5
It may not be immediately obvious, but the remainder of (the sum of the remainders) when divided by 6 is the remainder of the number when divided by 6.
So, (1+2)/6 has a remainder of 3.
(2+3)/6 has a remainder of 5.
(3+5)/6 has a remainder of 2.
(5+2)/6 has a remainder of 1.
. . .
(3+2)/6 has a remainder of 5.
-- I’m a certified WyzAnt e-tutor.

David W.
5 1 0 1 1 2 3 5 2 1 3 4 1 5 0 5 5 4 3 1
4 5 3 2 5 1 0 1 1 2 3 5 2 1 3 4 1 5 0 5
5 4 3 1 4 5 3 2 5 1 0 1 1 2 3 5 2 1 3 4
1 5 0 5 5 4 3 1 4 5 3 2 5 1 0 1 1 2 3 5
2 1 3 4 1 5 0 5 5 4 3 1 4 5 3 2 5 1 0 1
1 2 3 5 2 1 3 4 1 5 0 5 5 4 3 1 4 5 3 2
5 1 0 1 1 2 3 5 2 1 3 4 1 5 0 5 5 4 3 1
4 5 3 2 5 1 0 1 1 2 3 5 2 1 3 4 1 5 0 5
5 4 3 1 4 5 3 2 5 1 0 1 1 2 3 5 2 1 3 4
1 5 0 5 5 4 3 1 4 5 3 2 5 1 0 1 1 2 3 5
2 1 3 4 1 5 0 5 5 4 3 1 4 5 3 2 5 1 0 1
1 2 3 5 2 1 3 4 1 5 0 5 5 4 3 1 4 5 3 2
5 1 0 1 1 2 3 5 2 1 3 4 1 5 0 5 5 4 3 1
4 5 3 2 5 1 0 1 1 2 3 5 2 1 3 4 1 5 0 5
5 4 3 1 4 5 3 2 5 1 0 1 1 2 3 5 2 1 3 4
1 5 0 5 5 4 3 1 4 5 3 2 5 1 0 1 1 2 3 5
2 1 3 4 1 5 0 5 5 4 3 1 4 5 3 2 5 1 0 1
1 2 3 5 2 1 3 4 1 5 0 5 5 4 3 1 4 5 3 2
5 1 0 1 1 2 3 5 2 1 3 4 1 5 0 5 5 4 3 1
4 5 3 2 5 1 0 1 1 2 3 5 2 1 3 4 1 5 0 5
5 4 3 1 4 5 3 2 5 1 0 1 1 2 3 5 2 1 3 4
1 5 0 5 5 4 3 1 4 5 3 2 5 1 0 1 1 2 3 5
2 1 3 4 1 5 0 5 5 4 3 1 4 5 3 2 5 1 0 1
1 2 3 5 2 1 3 4 1 5 0 5 5 4 3 1 4 5 3 2
5 1 0 1 1 2 3 5 2 1 3 4 1 5 0 5 5 4 3 1
4 5 3 2 5 1 0 1 1 2 3 5 2 1 3 4 1 5 0 5
5 4 3 1 4 5 3 2 5 1 0 1 1 2 3 5 2 1 3 4
1 5 0 5 5 4 3 1 4 5 3 2 5 1 0 1 1 2 3 5
2 1 3 4 1 5 0 5 5 4 3 1 4 5 3 2 5 1 0 1
1 2 3 5 2 1 3 4 1 5 0 5 5 4 3 1 4 5 3 2
5 1 0 1 1 2 3 5 2 1 3 4 1 5 0 5 5 4 3 1
4 5 3 2 5 1 0 1 1 2 3 5 2 1 3 4 1 5 0 5
5 4 3 1 4 5 3 2 5 1 0 1 1 2 3 5 2 1 3 4
1 5 0 5 5 4 3 1 4 5 3 2 5 1 0 1 1 2 3 5
2 1 3 4 1 5 0 5 5 4 3 1 4 5 3 2 5 1 0 1
1 2 3 5 2 1 3 4 1 5 0 5 5 4 3 1 4 5 3 2
5 1 0 1 1 2 3 5 2 1 3 4 1 5 0 5 5 4 3 1
4 5 3 2 5 1 0 1 1 2 3 5 2 1 3 4 1 5 0 5
5 4 3 1 4 5 3 2 5 1 0 1 1 2 3 5 2 1 3 4
1 5 0 5 5 4 3 1 4 5 3 2 5 1 0 1 1 2 3 5
2 1 3 4 1 5 0 5 5 4 3 1 4 5 3 2 5 1 0 1
1 2 3 5 2 1 3 4 1 5 0 5 5 4 3 1 4 5 3 2
5 1 0 1 1 2 3 5 2 1 3 4 1 5 0 5 5 4 3 1
4 5 3 2 5 1 0 1 1 2 3 5 2 1 3 4 1 5 0 5
5 4 3 1 4 5 3 2 5 1 0 1 1 2 3 5 2 1 3 4
1 5 0 5 5 4 3 1 4 5 3 2 5 1 0 1 1 2 3 5
2 1 3 4 1 5 0 5 5 4 3 1 4 5 3 2 5 1 0 1
1 2 3 5 2 1 3 4 1 5 0 5 5 4 3 1 4 5 3 2
5 1 0 1 1 2 3 5 2 1 3 4 1 5 0 5 5 4 3 1
04/21/17

David W.
1 2 3 5 2 1 3 4 1 5
0 5 5 4 3 1 4 5 3 2
5 1 0 1
The 1000th remainder in the list is 1. The remainder of ( T1000 / 6 ) is 1.
04/21/17

David W.
1 2 3 5 2 1 3 4 1 5
0 5 5 4 3 1 4 5 3 2
5 1 0 1
The 21st remainder in the list is 5. The remainder of ( T21 / 6 ) is 5.
The 1000th remainder in the list is 1. The remainder of ( T1000 / 6 ) is 1.
04/21/17

Katie B.
04/21/17

David W.
04/22/17

Katie B.
04/23/17
Adi X.
04/21/17