Math Pattern: Find the formula (ie the nth term) for the sequence 0,0,1,3,6,10…

Answer

0,0,1,3,6,10

Let A(n) denote the (n + 2)th term in the sequence. We “choose” ( n + 2)th term because we start counting from
‘-1’ because we want to “manipulate beautifully”

0 = 0 + 0 n = -1
1 = 0 + 1 n = 0
3 = 1 + 2 n = 1
6 = 3 + 3 n = 2
10 = 6 + 4 n = 3
….. See the pattern?
A(n) = A(n – 1) + n where n = -1,0,1,2 etc.

So,

A(0) – A(-1) = 1
A(1) – A(0) = 2
A(2) – A(1) = 3
A(3) – A(2) = 4
+ ….
A(n -1) – A (n – 2) = (n – 1)
A(n) – A( n – 1) = n

“Adding” the above… observe “the cancellation of the left hand side of the equation”…

A(n) – A(-1) = 1 + 2 + … + n = n*(n + 1) /2
A(n) = A(-1) + n*(n + 1) /2 but A(-1) = 0
A(n) = n(n +1)/2 = ( n + 2)th term with n= -1, 0, 1, 2 etc.

Suppose you wanted to “start counting with n = 1,2” . Can we derive a “formula for the nth term to mean A(n)?” Yes. We proceed as follows :

Just change the variable.
use.
A(y) = (y – 1) ( y – 2) / 2 for y = 1,2 etc.

or,

A(n) = (n – 1) ( n – 2) / 2 = nth term ; for n = 1,2 etc
Done .