Can someone please help me find an explicit formula for the sequence 1,2,4,7,11,16,22... ?

Question

tkhunny · Answer

Subtract each value after the first from the number that preceeds it.

2 - 1 = ??
4 - 2 = ??
7 - 4 = ??

Do them all and see if you notice a pattern.

anonymous · Answer

Yurps Nice job didnt get it at 1st

anonymous · Answer

I have already got that far. I know the pattern increases by one more than the first each time. I have even made the recursive formula for it (I think). 
$$a _{n}= a _{n-1} + n$$But I do not know how to make the explicit formula. I am repeatedly getting stuck.

anonymous · Answer

Wait, no. I don't think that formula is right either...

anonymous · Answer

Is something wrong with my sequence? is it even formula-able? O.o

anonymous · Answer

I'll check back... Hopefully someone can help me!

tkhunny · Answer

You must learn DISCOVERY.  Try some things.

$a_{n} = n^{2}$

1
4
9
16
...

Okay, that's a little too fast.

$a_{n} = n^{2} - n$

0
2
6
12
...

Off by at least 1 at the beginning

$a_{n} = n^{2} - n + 1$

1
3
7
13
...

I see it.

$a_{n} = ½(n^{2} - n + 2)$

It takes a little effort.

Note: that little experiment at the beginning told us that a quadratic function would be sufficient.  We didn't just do that for fun.

anonymous · Answer

Thank you, truly. That really helped me understand. :)