Having ALOT of trouble with Sequences, Series and Combinatorics (is this normal?)

[LG154]

Hey folks, I've just been working through grade 12 pre-calculus as an upgrading student and had been challenged quite alot by this level in comparison to grade 11 pre-calculus. That said, I've managed to get through it relatively alright (as in, I've been able to complete my assignments by mostly referring to khan and online algebra calculators to help understand the concept and check my work), but the unit on sequences, series and combinatorics is on a whole other level for me.

I've essentially worked my way through understanding arithmetic and geometric sequences, but my assignment expects me to figure out sequences that are neither of those things, and I can't find any references online on how to start with these.

For example: "Find the general term in simplest form for the sequence: 2, 1, -4, 7, -10, 13, -16
I was told that there were no available formulas and i was to just experiment with it until i got something.

When looking through the text, I was able to find a relatable sequence that helped make sense of the initial 2 positives and alternating - +'s.
The example being: a(n)=-1^n-1(3n-5)

But, when trying to reverse engineer this into something that'd fit my question I'm completely stumped as I can't find anything that'll give me that jump from 1 to 3... The |3| added value is there, but this is just one example of questions that seem to be coming out of the left-field in relation to the rest of the work and reading i've done leading up to this point. I think the problem is that I've been depending on formulas that the book gave me (Mainly have just been doing graphing, function, models, equations, inequalities, etc. up to this point which I can't relate to any of this stuff)

Does anyone have any suggestions for material to read up on that'll help me with the logic behind these kinds of questions?

Kind of a mouthful, but I'd appreciate any input.

Thanks!

 

[HallsofIvy]

For example: "Find the general term in simplest form for the sequence: 2, 1, -4, 7, -10, 13, -16
I was told that there were no available formulas and i was to just experiment with it until i got something.

Mathematics is not just formulas. It is largely looking for "patterns". Further, a "sequence" is just a list of numbers that might, conceivably, be completely random and have NO "general term". Given such a list I might see that going from "2" to "1" we go down 1 then down 5 more to -4 then up 11 to 7 but that doesn't give any pattern. Multiplying doesn't give anything either. 1 is 1/2 of 2 but -4 is -4 times 1, 7 is -7/4 times 4- no pattern there. There doesn't appear to be anything to do with squares or square root, cubes or cube roots, etc. I don't see any reasonable pattern here.

One thing I will say, given "n" (x, y) data points, there exist a polynomial, y= p(x), of degree n-1 that passes through those data points. Here thinking of this sequence as \(\displaystyle a_1= 2\), \(\displaystyle a_2= 1\) etc., and then as data points (1, 2), (2, 1), (3, -4), (4, 7), (5, -10), (6, 13), (7, 16) there exist a sixth degree polynomial such that \(\displaystyle p(n)= a_n\) but I doubt that is what is wanted here. The fact is that there are infinitely many different sequences that have the same first 7 values.

When looking through the text, I was able to find a relatable sequence that helped make sense of the initial 2 positives and alternating - +'s.
The example being: a(n)=-1^n-1(3n-5)

But, when trying to reverse engineer this into something that'd fit my question I'm completely stumped as I can't find anything that'll give me that jump from 1 to 3... The |3| added value is there, but this is just one example of questions that seem to be coming out of the left-field in relation to the rest of the work and reading i've done leading up to this point. I think the problem is that I've been depending on formulas that the book gave me (Mainly have just been doing graphing, function, models, equations, inequalities, etc. up to this point which I can't relate to any of this stuff)

Does anyone have any suggestions for material to read up on that'll help me with the logic behind these kinds of questions?

Kind of a mouthful, but I'd appreciate any input.

Thanks!

 

[Romsek]

Yes it's perfectly normal. Series in calc II was where a fair number of students decided the engineering track wasn't for them. Combinatorics was usually a little further along but was very good in its own right at producing a classroom of blank stares.

Welcome to the club!

 

[Steven G]

Yes it's perfectly normal. Series in calc II was where a fair number of students decided the engineering track wasn't for them. Combinatorics was usually a little further along but was very good in its own right at producing a classroom of blank stares.

Welcome to the club!

Really? In my college the students seemed to do ok with series. I was the only math major in the class with quite a few engineering majors. There were just a few of us crazy ones who decided to study math!

 

[Steven G]

I too do not see a pattern. If all else fails, the method Prof Halls mentioned will always work.

 

[Dr.Peterson]

Yes it's perfectly normal. Series in calc II was where a fair number of students decided the engineering track wasn't for them. Combinatorics was usually a little further along but was very good in its own right at producing a classroom of blank stares.

This is not calc II, but precalc (though I have not idea what grade 11 and grade 12 precalc would be). It should not be nearly as hard.

On the other hand, if an exercise asks to determine the formula for a sequence from a few terms, then it is an invalid problem, and of course students would have trouble.

If I were helping face to face, I would take the book in hand and check what is being taught in this section (as well as confirm the wording of the problem). My hope is that, at the least, some example has been given of a sequence other than arithmetic or geometric, and that there is some indication that a certain type of sequence is expected.

As for the specific problem, I do see a pattern, though is it a sort of hybrid. 2, 1, -4, 7, -10, 13, -16. My first thought was that if we drop the signs, we get 2, 1, 4, 7, 10, 13, 16, which would be an arithmetic sequence if only the first term were changed to -2; so maybe there was a misprint. But then I realized that if we change the sign of every odd term, we get -2, 1, 4, 7, 10, 13, 16, which is arithmetic. So we can start with this sequence, a(n) = 3n - 5, and then alternate the signs.

So the OP's idea (after a correction) is right: a(n)=(-1)^{n-1}(3n-5). (Notice the added parentheses so that (-1) is the base, not just 1, and the added braces, so that n-1 is the exponent, not just n.) This is correct except that the signs are reversed; we want the first term and every odd term to be negated, so we need a(n)=(-1)^{n}(3n-5).

But I'm confused. There is no "jump from 1 to 3" in this sequence, so I'm wondering if that was the example given, and we haven't been shown the one the OP can't do.

In any case, in the apparent context of having learned about arithmetic and geometric sequences, and having seen one or two modifications of them (such as this, changing signs), the thing to do is to look for those particular kinds of sequences. Without such a context, as I said, the problem is entirely invalid (though all to common in some curricula). With it, we can restrict our thinking to a few possibilities.

 

[LG154]

Thanks for the replies guys! I've just been away from home on a surprise trip, but have found another example of a question like it that was sent by my teacher in response to my questioning:

https://imgur.com/kWroK74

I suppose what's the most frustrating to me about all of this is that I feel like there is a big gap in the knowledge leading up to this from what i've been studying so far. I essentially did a bunch of graphing and function exercises leading up to this and feel like there isn't any frame of reference for what I'm doing now, despite re-reading the chapter and going over all of the videos that i can find online.

What more is that I actually felt quite comfortable with the material that i've been working on leading up to this point.

Just found my study outline (turns out i brought it with me). The outline looks something like this:
-graphs,functions, models, equations, inequalities
-exponential functions and logarithmic functions
-trigonometric functions
-trigonometric identities, inverse functions, and equations
-applications of trigonometry, systems of equations and matrices
-conic sections

and then...*grrr*

-Sequences, series and combinatorics.

Is there anything here that stands out to anyone that i should go over to make more sense out of?

I'm currently reading out of the beecher penna bittinger algebra and trigonometry 5th edition.

 

[Dr.Peterson]

This example is just the negative of the other, since the exponent on -1 is off by 1. And it appears that it just asks you to evaluate the expression (assuming what you show is the solution -- you didn't show the problem itself). If so, then it's not really the same kind of problem at all.

What I don't know is what in the original question was an example you were given, and what is the problem you are asking about.

The list you gave here doesn't help much; it's just an overall outline of general topics, none of which are unusual. What I'd want to look at is the chapter and section you are in! At least please show the exact wording of some problems with their full instructions, so we can tell you what it is telling you to do.

But overall, these appear to be more or less just "recognition" problems -- look for alternating signs, undo them, and expect an arithmetic or geometric sequence. Nothing harder than that, as far as I can see. It's just unfamiliar to you.

 

[LG154]

yeah that's right! it's just the unfamiliarity that's been bugging me. There appears to be this recognition element that is a bit novel from the stuff that I was used to doing up till now and I've just to find some kind of resource that gives a good run through of examples of all of these variations.

For example: the (-1)^n-1 being applied when you see alternating +s and -s in some sequences makes sense, 3^n is what you apply when you see a sequence that goes from 3,9,27,81... or when you see 3,-9,27,-81... apply (-1)^n-1 * 3^n also makes sense.

I just know that there's a good amount of different situations like this and i was wondering if there was some way to get a good introduction on the various patterns that are commonly seen so that I can sort of work it out in my head how to observe certain patterns and how to tackle them.

Am i just severely lacking in general algebra to not be able to come up with some of these patterns or are teachers usually pretty generous with examples when introducing sequences and series.

 

[LG154]

Sorry, in regards to that example. What the fellow meant was that for the sequence of numbers along the right hand side, I was to figure out the formula. The -2, -1, 4, 7.... being the only information in which i was to find the general term (which is provided in the top). I originally couldn't make out the - to + to -... which prompted the teacher to explain I was explained that the (-1)^n-1, which i now know is a general function for such sequences.

 

[Dr.Peterson]

I can't say any more without seeing what the entire problem set looks like. They may be giving you only certain types of sequences, or they may start off with patterns explicitly shown that you are expected to recognize, like [MATH]\frac{1}{3}, \frac{2}{4}, \frac{3}{5}, \dots[/MATH], which "clearly" is [MATH]a(n) = \frac{n}{n+2}[/MATH]. If this had been given as [MATH]\frac{1}{3}, \frac{1}{2}, \frac{3}{5}, \dots[/MATH], it would be much trickier, and I would object to it in this context.

What skills you need to develop depends on exactly what sort of problems are being given, and what assumptions you are allowed to make.

 

[LG154]

Luckily, my inlaws happened to be pretty great at math and helped tutor me a bit through it. After going through some questions step by step with them, the concepts of creating equations clicked for me and I was able to proceed with references in the text to work out questions in my assignment.

It seems like it's one of those things that I had to really work out in person with someone to understand.

Thanks everyone!