Pattern: 10, 17, 26,
what will the most likely number in this pattern be greater than 350?

  362 is the 17th term of this pattern

Step-by-step explanation:

First differences of the sequence are ...

  17-10 = 7, 26-17 = 9, 37 -26 = 11

And second differences are ...

  9-7 = 2, 11-9 = 2

The constant second differences tell you the pattern is described by a second degree polynomial. That polynomial can be written as ...

  an^2 +bn +c . . . for term number n

Filling in the first 3 sequence values, we get ...

  a + b + c = 10

  4a +2b +c = 17

  9a +3b +c = 26

___

Any of several means can be used to solve these.

We can subtract the first equation from the other two to get ...

  3a +b = 7 . . . . . . . . [eq4]

  8a +2b = 16 . . . . . . [eq5]

Subtracting [eq4] from half of [eq5], we get ...

  (4a +b) -(3a +b) = 8 -7

  a = 1

Putting this value into [eq4] gives ...

  3 +b = 7

  b = 4

and using these values for a and b in the first equation gives ...

  1 + 4 + c = 10

  c = 5

So the pattern is described by ...

  an = n^2 +4n +5

We want to find the term that is more than 350, so ...

  350 < n^2 +4n +5

  349 < n^2 +4n +4 . . . . subtract 1 to make a square on the right

  349 < (n +2)^2

  √349 ≈ 18.68 < n +2 . . . . take the positive square root

  16.68 < n . . . . .  . . . . . . subtract 2

The first term greater than 350 is the 17th term. It is 362.

In the process of "completing the square", we observe that the pattern rule can be written ...

  an = (n+2)^2 +1

step-by-step explanation: