What is the pattern use words not numbers

Question

The numbers are 1,3,6,10

Answer 1

A Sequence is a list of things (usually numbers) that are in order.  This problem lists the numbers 1, 3, 6, 10.
 
The terms in a sequence can be numbered (PLZ start at 1, although some people start with term 0).  So, term 1 is 1, term 2 is 3, term 3 is 6, and term 4 is 10.
 
There is often a rule that allows you to find each term either using the term number or the value of previous term(s).  To find that rule, determine how the numbers change.  Start by subtracting terms, because that's a really easy sequence (like 2, 4, 6, 8 has a difference always equal to 2).  Here, the differences are 2, 3, 4  (well, that's a pretty easy one also).  The next number could be 15.
 
Let's express the rule.
  For term 1, the value is 1.
  For term 2, the value is 3, which is the term number (2) plus the previous term value (1)
  For term 3, the value is 6, which is the term number (3) plus the previous term value (3)
  For term 4, the value is 10, which is the term number (4) plus the previous term value (4)
 
Each term is the sum of the term numbers (not the term values) so far.
 
So, given a term number, what is the term value?  We'd like to express the rule so that we don't have to find all the other terms back down to 1, so we use Gauss' formula for the sum of the numbers from 1 to n,  which is n(n+1)/2.
 
For term n, the value is the sum of the term numbers (not the term values) from 1 up to the term number [ n(n+1)/2 ].
Try it:
   term    value
     1          (1(1+1)/2) = 1
     2          (2(2+1)/2) = 3
     3          (3(3+1)/2) = 6
     4          (4(4+1)/2) = 10
 
So, it looks like the rule is:   Tn = n(n+1)/2
 
And, if the problem asks, "What is the value of term 23?", we would say, 23*24/2 = 276.
 
Note:  this formula is very important and often appears in problems.

Answer 2

Jazmin, the pattern should have begun with zero.  It would have made more sense. 

Each time you are adding one more to get to the next number.  One plus two equals three, three plus three equals six, six plus four equals ten, and the next number in the series would be fifteen because ten plus five is fifteen.

 

Does this make more sense to you?

Answer 3

It looks like the gap between numbers increases by 1 each time.

So, the gap between 1 and 3 is 2.

The gap between 3 and 6 is 3

and the gap between 6 and 10 is 4