- #1
Robb
- 225
- 8
- Homework Statement
- Let ##S_n##be the hexagonal arrangements consisting of n rings of dots for ##n \in {1, 2, 3}##. Let ##a_n## be the number of dots in ##S_n##. Find formulas for ##a_n## and ##\sum_{k=1}^n a_k##.
- Relevant Equations
- ##a_1=1, a_2=7, a_3=19##
##a_n = a_{n-1} + 6(n-1)##
##=a_{n-2} + 6(n-1) + 6(n-2)##
##=a_{n-3} + 6(n-1) + 6(n-2) + 6(n-3)##
##\vdots##
##=1 + 6 [(n-1) + (n-2) + (n-3) + \cdots + 1]##
##= 1 + 3n(n-1)##
I'm not sure how to get to the last line from the second to the last line. Please advise. Thanks!
##=a_{n-2} + 6(n-1) + 6(n-2)##
##=a_{n-3} + 6(n-1) + 6(n-2) + 6(n-3)##
##\vdots##
##=1 + 6 [(n-1) + (n-2) + (n-3) + \cdots + 1]##
##= 1 + 3n(n-1)##
I'm not sure how to get to the last line from the second to the last line. Please advise. Thanks!