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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.

 Mar 5, 2019
 #1
avatar+101804 
+1

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

 

 

 

 

 

cool cool cool

 Mar 5, 2019
 #2
avatar+101804 
+1

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

 

 

cool cool cool

 Mar 6, 2019
edited by CPhill  Mar 6, 2019
edited by CPhill  Mar 6, 2019

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