The first term of a sequence is 13. Starting with the second term, each term is the sum of the cubes of the digits in the previous term. For example, the second term is \(1^3 + 3^3 = 28\). Find the 100th term.
The sequence starting 1, 5, 12, 22, 35, counts the number of points in each diagram. Find the number of points in the tenth diagram.
+128475 Second one......can be solved with a sum of differences
1 5 12 22 35
4 7 10 13
3 3 3
We have 2 non-ero rows....so.... will have a 2nd power polynomial ax^2 + bx + c
a + b + c = 1 (1)
4a + 2b + c = 5 (2)
9a + 3b + c = 12 (3)
Subtract (1) from (2) and (1) from (3)
3a + b = 4 ⇒ -6a - 2b = -8 (4)
8a +2b = 11 (5)
Add (4) and (5)
2a = 3
a = 3/2
3(3/2) + b = 4
9/2 + b =4
b = 4 = 9/2
b = -1/2
3/2 - (1/2) + c = 1
1 + c = 1
c = 0
So....the generating function is
(3/2)x^2 - (1/2)x
So...the 100th term is
(3/2)(100)^2 - (1/2)(100) =
14,950
+128475 1. This seems that, at first glance, it would lead to huge numbers...it doesn't....
Note the pattern .......
Term
1 13
2 1^3 + 3^3 = 28
3 2^3 + 8^3 = 520
4 5^3 + 2^3 + 0^3 = 133
5 1^3 + 3^3 + 3^3 = 55
6 5^3 + 5^3 = 250
7 2^3 + 5^3 + 0^3 = 133
8 1^3 + 3^3 + 3^3 = 55
9 5^3 + 5^3 = 250
Note
4mod 3 = 1 = 1st number in the repeating cycle = 133
5 mod 3 = 2 = 2nd number in the repeating cycle = 55
6 mod 3 = 0 = 3rd number in the repeating cycle = 250
So..for the 100th term.....
100 mod 3 = 1 = the 1st number in the repeating pattern = 133
