   SEARCH HOME Math Central Quandaries & Queries  Question from danii, a student: im trying to solve the nth term for this pattern. 1   3   6   10   15   21   28 any help would be appretiated Hi Dani,

Let's write out a table for "n" and the value S(n):

 n S(n) 1 1 2 S(1)+2 3 S(2)+3 4 S(3)+4 5 S(4)+5 ... n S(n-1)+n

So you could say that the nth term is just the (n-1)th term plus n. But this may not be what you are seeking. Let's write it another way:

 n S(n) 1 1 2 1+2 3 1+2+3 4 1+2+3+4 5 1+2+3+4+5 ... n 1+2+...+(n-1)+n

So really, S(n) is the sum of the first n natural numbers.

Can you think of a way to express the sum of the first n natural numbers in an easy formula?

Here's a hint:

 S(6) = (1+2+3+4+5+6) = (1+2+3+4+5+6)×2×½ = (1+2+3+4+5+6+1+2+3+4+5+6)×½ = (1+6+2+5+3+4+4+3+5+2+6+1)×½ = ((1+6)+(2+5)+(3+4)+(4+3)+(5+2)+(6+1))×½ = ((6+1)×6)×½ = 21 (as we expected.

Cheers,
Stephen La Rocque.     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.