Basic Math/pattern

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Question
I'm trying to find the pattern here. What would the number possibly be in place of "x"?

7, 19, X, 4.

Can the number be within the range of 1-26?

Thanks!

Answer
Hello,

Problems like these are questionable without some kind of idea or theme.
The reason is that there are infinitely many "correct" answers (usually
only one "simplest" answer though).

I'll use this problem as an example.  Since we are given 3 values, I assumed
the sequence has a form f(n)=a*n^2 + b*n +c, where a,b, and c are constants
with f(1)=7, f(2)=19, f(3)=x=?, and f(4)=4.  Using the known values, we get 3
equations for a, b, and c:
f(1)=7 gives: a+b+c=7, f(2)=19 gives: 4a+2b+c=19, and f(4)=4 gives: 16a+4b+c=4.

Solving these three equations for a, b, and c gives: a=-13/2, b=63/2, and c=-18.
Thus, one possible solution is: f(n)=(-13/2)n^2+(63/2)n-18 = (1/2)(63n-13n^2-36).
Now use this to determine f(3)...f(3)=(1/2)(189-117-36)=18.  So 18 is a fine answer!

However, what if we use a different model, say f(n)=a*n+b/n+c.  This model gives
a=-14, b=-52, and c=73 to get: f(n)=73-14n-52/n.  Now let n=3 to get f(3)=41/3, which
is also "a correct answer."  See what I mean?

Probably, the person who made the problem had the first way in mind, so f(3)=18.
But, who knows??!!!!

A. Mantell

Basic Math

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Abe Mantell

Expertise

Hello, I am a college professor of mathematics and regularly teach all levels from elementary mathematics through differential equations, and would be happy to assist anyone with such questions!

Experience

Over 15 years teaching at the college level.

Organizations belong to
NCTM, NYSMATYC, AMATYC, MAA, NYSUT, AFT.

Education/Credentials
B.S. in Mathematics from Rensselaer Polytechnic Institute
M.S. (and A.B.D.) in Applied Mathematics from SUNY @ Stony Brook

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