Find the first 40 terms of the sequence defined by an+1= (1)/(2) an if an is an even numb

Find the first 40 terms of the sequence defined by an+1=...

Find the first 40 terms of the sequence defined by $$ a_{n+1}= \begin{cases}\frac{1}{2} a_n & \text { if } a_n \text { is an even number } \\ 3 a_n+1 & \text { if } a_n \text { is an odd number }\end{cases} $$ and $a_1=11$. Do the same if $a_1=25$. Make a conjecture about this type of sequence.

Find the first 40 terms of the sequence defined by

$$
a_{n+1}= \begin{cases}\frac{1}{2} a_n & \text { if } a_n \text { is an even number } \\ 3 a_n+1 & \text { if } a_n \text { is an odd number }\end{cases}
$$

and $a_1=11$. Do the same if $a_1=25$. Make a conjecture about this type of sequence.

Key Concepts

Recursive Sequences

Recursive sequences are those in which each term is defined in relation to one or more previous terms. They allow the generation of a series of numbers by repeatedly applying the same rule, and they play an important role in understanding patterns and long-term behavior in mathematics.

Recurrence Relations

Recurrence relations are equations that express each term in a sequence as a function of preceding terms. They are fundamental in analyzing and solving sequences, as they provide a systematic way to describe the evolution of a sequence over time.

Parity Considerations in Recurrence

In many recursive sequences, including those of a conditional nature, the behavior of the sequence can change based on whether a number is even or odd. Recognizing and handling parity is crucial as it often determines which branch of a piecewise defined recurrence relation to apply.

Collatz Conjecture

The Collatz Conjecture is a famous unsolved problem in mathematics concerning a specific recursive sequence where each term’s definition depends on its parity. It conjectures that no matter which positive integer starts the sequence, repeated application of the rule will eventually reach the cyclic pattern, typically involving 1, despite the seemingly unpredictable intermediate behavior.

Transcript

00:02 Find the first 40 terms of the sequence given by the following formula.

00:06 In case 1, let's look at a1 equals 11.

00:10 A2.

00:12 This will be 3a1 plus 1, since a1 is an odd number, excuse me.

00:19 When it's an odd number, you use this formula, 11 was odd.

00:23 So we use this formula here, and that's how we got a2.

00:29 So that's kind of the reasoning there.

00:31 The arrows indicate the reasoning to find the following values of the sequence.

00:37 So this is just 3 times 11 plus 1, 34.

00:42 Now a3, since 34 is an even number, we divide it by 2.

00:50 34 even.

00:52 Use this formula and then that brings us to why we use this formula over here.

00:58 And i won't spell that out every time, but if it's an even or odd number, that tells me which formula to use because it's piecewise function.

01:09 A4, that's 3 times 17 plus 1 so 52 a 5 well we have an even number so we'll divide that by 2 we have another even so we'll also divide that by 2 that's odd multiply by 3 add the 1 8 that's even so divided by 2 same thing for a 9 also a 10 divided by 2 we have another odd so other formula and then even again even even still even and then finally we have a one not finally we stop more terms but we we've reached one and now we'll see that it's well it'll start repeating and a following pattern so if we use the formula for this odd number we get three times one plus one which is four but then once we get back to four it will just continue to cycle four to one 421 and so on.

02:27 And here we see the pattern once again, 421, and this will repeat.

02:33 So you can continue just writing those 421, 421s, and then the last four terms of the sequence, because we are going up to 40 terms, that's what's being asked.

02:45 And the final term should be a 4.

02:49 So this is the sequence, the first 40 terms in this first case here, a1 is 11...

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