Functions and Sequences

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Problems about functions and sequences.

A sequence is a function with a domain that is a subset of the $\answer [format=string]{integers}$ .

An arithmetic sequence has a constant $\answer [format=string]{difference}$ between consecutive terms.

An arithmetic sequence is a(n) $\answer [format=string]{linear}$ function.

An geometric sequence has a constant $\answer [format=string]{ratio}$ between consecutive terms.

A geometric sequence is a(n) $\answer [format=string]{exponential}$ function.

Consider a sequence that begins $9, 12$ .

(a): Continue the sequence so that it is an arithmetic sequence: $9, 12, \answer {15}, \answer {18}, \answer {21}, \answer {24}, \dots$
(b): Assuming that $f(1)=9$ , write a recursive formula for the sequence: $f(n+1)=\answer {f(n)+3}\textrm {, for }n\ge \answer {1}.$
(c): Assuming that $f(1)=9$ , write an explicit formula for the sequence: $f(n)=\answer {3n+6}\textrm {, for }n\ge \answer {1}$

Consider a sequence that begins $9, 12$ .

(a): Continue the sequence so that it is an geometric sequence: $9, 12, \answer {16}, \answer {9(4/3)^3}, \answer {9(4/3)^4}, \answer {9(4/3)^5}, \dots$
(b): Assuming that $f(1)=9$ , write a recursive formula for the sequence: $f(n+1)=\answer {f(n)(4/3)}\textrm {, for }n\ge \answer {1}.$
(c): Assuming that $f(1)=9$ , write an explicit formula for the sequence: $f(n)=\answer {9(4/3)^{n-1}}\textrm {, for }n\ge \answer {1}$

The $3n+1$ problem involves integer sequences defined as follows: Start with any positive integer. Then, if the current term is even, the next term is one half of the current term. If the current term is odd, the next term is $3$ times the current term plus $1$ .

Suppose the first term is 3. Find the next several terms.

$3, \answer {10}, \answer {5}, \answer {16}, \answer {8}, \answer {4}, \answer {2}, \answer {1}, \answer {4}, \answer {2}, \answer {1}.$

Note that once the sequence reaches $1$ , it cycles back to 1 repeatedly.

If we use function notation to describe the sequence, then the recursive rule may be stated as follows:

If $f(k)$ is even, then $f(k+1) = \answer {f(k)/2}$ .
If $f(k)$ is odd, then $f(k+1) = \answer {3f(k) + 1}$ .

And given $f(1)=3$ , it follows that $f(2)=\answer {10}$ , $f(3)=\answer {5}$ , and so on.

Is there an explicit formula for this function, $f$ ?

Yes, it is linear. Yes, it is quadratic. Yes, it is exponential. Yes, it is trigonometric. Probably not.

Claim: No matter the initial (positive integer) value, the sequence will always reach 1.

This claim is often called the Collatz Conjecture, in honor of Lothar Collatz, who introduced the idea in 1937. As of 2020, the conjecture has been checked by computer for all starting values up to $2^{68} \approx 2.95\times 10^{20}$ . The statement is still a conjecture (rather than a theorem) because no one has been able to prove that it holds for all possible initial values. For more information, see Wikipedia.

The following table shows historical data collected by the National Weather Service.

Let $F(t)$ denote the actual or predicted temperature (in degrees Fahrenheit) $t$ hours after the start (at midnight) of November 6, 2021.

(a): Note that the hourly readings all occur at $53$ minutes after the hour. For ease of entry, we are going to take 53 minutes as 0.9 hours (though, in fact, 0.9 hours is exactly $\answer {54}$ minutes). This minor simplification, allows us to write, for example, $F(0.9) = \answer {36}\text { and }F(12.9)=\answer {52}.$
(b): Negative values of time occur on November $\answer {5}$ or earlier, counting hours backwards from midnight. For example, $F(-1.1) = \answer {35}\text { and }F(-7.1)=\answer {50}.$
(c): From real-world data, it is often reasonable to estimate values that are not directly available. In particular, to interpolate is to estimate between available data values, and to extrapolate is to estimate beyond available values. From this data, for example, we can estimate the following: $F(9.2)=\answer {32}, F(9.6)=\answer {35}, F(-4.6)=\answer {44}, F(14.5)=\answer {54}\text {, and }F(17)=\answer [tolerance=1]{56}.$

From this table, is it reasonable to estimate $F(30)$ ?

Yes, use the formula. Yes, just continue the trend. Probably not.

With this small data set, it is not reasonable to extrapolate more than a few hours beyond the available data. Meteorologists (weather forecasters) use enormous data sets about nearby and historical weather to develop sophisticated models that are usually pretty accurate a few days into the future and somewhat accurate up to ten days.

Is there an explicit formula for this function, $F$ ?

Yes, it is linear. Yes, it is quadratic. Yes, it is exponential. Yes, it is trigonometric. Probably not.