Answer:
f(n) = 5n(n -1)/2 +9
Step-by-step explanation:
The rule will depend on what kind of sequence you think this is. An infinite number of rules can be dreamed up that will give these three values.
The simplest polynomial rule is one of 2nd degree. We know this because the first differences are 14-9 = 5 and 24-14 = 10, so are not constant. Only three values are given, so only a polynomial of degree 3-1=2 can be specified by these points.
If we say the polynomial is ...
f(n) = an^2 +bn +c
Then the three given points give rise to three equations:
9 = a + b + c
14 = 4a +2b +c
24 = 9a +3b +c
Solving these by your favorite method gives ...
a = 5/2, b = -5/2, c = 9
So an explicit polynomial rule can be written as ...
f(n) = 5n(n -1)/2 +9
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Additional comment
A sequence such as this one can also be described by an exponential rule. Here's one that describes these points:
f(n) = 5·2^(n-1) +4
The fact that the first differences (5, 10) are related by a factor of 2 suggests 2 as the base of the exponent.
