Find a formula for the sequence 1,3,6,10,15,...

Question

Write a formula/formulae for the following sequence:

b). 1,3,6,10,15,...

I am not getting any pattern here, from which to derive a formula. This sequence does not look like the examples I could solve: like

a) 1,0,1,0,1...

where I got that $S_n =1 $ (for $n=1,3,5,7,...$) and $S_n=0$ (for $n=2,4,6,8,...$)

or

c.) 1,1,1,2,1,3,1,4,1,...

where $S_n=1$ for $n=1,2,3,5,7,...$ and $S_n= n/2$ for $n=4,6,8,.. $

Answer 1

For (b) you can observe the difference between next and previous terms is in A.P.

Thus you can define it recursively as $s_1 = 1$, $s_{n+1}=s_n + (n+1)$.

Answer 2

Mike Pierce has already given the answer.

Still let me mention that the $n$-th term of the sequence is the sum of the first $n$ natural numbers.

Hence $$s_n = \frac{n(n+1)}{2}$$

It is also possible to derive the above by solving the recurrence relation given by Amey Deshpande.