Answer:
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
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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.
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In the process of "completing the square", we observe that the pattern rule can be written ...
an = (n+2)^2 +1
