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### Course: Algebra 1 (Eureka Math/EngageNY) > Unit 3

Lesson 3: Topic A: Lessons 1-3: Arithmetic sequence formulas- Explicit formulas for arithmetic sequences
- Explicit formulas for arithmetic sequences
- Explicit formulas for arithmetic sequences
- Arithmetic sequence problem
- Recursive formulas for arithmetic sequences
- Recursive formulas for arithmetic sequences
- Recursive formulas for arithmetic sequences
- Converting recursive & explicit forms of arithmetic sequences
- Converting recursive & explicit forms of arithmetic sequences
- Converting recursive & explicit forms of arithmetic sequences

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# Converting recursive & explicit forms of arithmetic sequences

Sal is given an arithmetic sequence in explicit form and he converts it to recursive form. Then he does so the other way around!

## Want to join the conversation?

- So an arithmetic sequence has a constant rate right(24 votes)
- Right, because the common difference is always constant in arithmetic sequences.(18 votes)

- Is arithmetic sequence basically a linear function with the domain of positive integers?

Are there other differences?(11 votes)- Late comment. The others are correct. To expand, if you graphed your sequence, you would get what looks like dots that can be connected by a line (just like the functions in the previous playlist). As we only look at positive integers, the line wouldn't actually be drawn.(6 votes)

- what's the meaning of "h" and "g" f is function and n stand for number of times (i guess) but what about the first two(6 votes)
- Each function has a name which is {h, g, f,...}, an input like n, an output {h(n), g(n), f(n),...}, and a relationship that connects the input and the output like the formula in this video:
`h(n) = -31 - 7(n-1)`

Hope this helps!(11 votes)

- so can the recursive formula be stated in 2 ways or is there a preferred version.

g1 = x, gn = g(n-1) + y

or g(n) = x if n = 1

= g(n-1) + y if n > 1(6 votes)- the recursive formula can be stated in two ways/ forms. however, there is the preferred version, which is g(n)= g(n-1) +y. technically you can change it into g(n)= y+ g(n-1). it's just easier to see/ visualize the function in the first format rather the second one.(6 votes)

- at1:20doesn't he mean subtract seven not negative seven(6 votes)
- Yeah, it seems he corrected this in the video annotations after the fact(3 votes)

- Why would you even want to do this? How is this used in real life and why do you need to change formulas?(5 votes)
- Q1: Sequences come in handy in higher maths when you begin calculus. In the "real world", sequences are used in many areas, including home loans and engineering (to name a few).

Q2: The changing of the formulas is shown for your knowledge. Since there are 2 formulas available, it's good to know how to get one from the other. Also, students often prefer one over the other. Given a formula, those students can convert it into their preferred one.(4 votes)

- In which part of daily life, this type of theory is used?(3 votes)
- Well, most likely you won't use this however algebra, and other difficult math are used to show how fast and capable you are of understanding hard concepts. So it you make A+'s in trigonometry or something like that the company you are trying to apply for will think "Ok, he can understand hard things and do well let's hire him!" Does that make any sense?(5 votes)

- There seem to be 2 ways of writing this. One has the n in braces (g(n)) and the other one has n as a small subscript of the sequence. Which one is preferable under which circumstances?(4 votes)
- g(n) is a function. You basically put any value bounded by constrains for n and you can get a value that the equation g(n) denotes. This is more general and used mostly for Explicit formulas.

The small subscript is a way to denote which term in the sequence (Starting from 1). For example F10 (Where 10 is the subscript) then this means the 10th term in the sequence F. This is more used in recursive.

But which to use is based your what you prefer and the problem. There really isn't a set of rule that constrains you from using which.(2 votes)

- He never explains what an explicit formula is. Or how to formate one.(2 votes)
- https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:sequences/x2f8bb11595b61c86:constructing-arithmetic-sequences/a/writing-explicit-formulas-for-arithmetic-sequences this article shows what explicit formula is its basically A+B(n-1) where A is the first term and B is the common difference and N is the nth term, what it does is EXPLICTICLLY solves for the sequence, you put in a nth term you output a explicit answer. :)(5 votes)

- Why did Sal say subtract -7. Wouldn't that be the same as seven?(2 votes)
- This is a known error in the video. A correction box pops up at about1:10in the video and tells you that Sal meant to say just "subtract 7". If you watch in full screen mode, you will not see the correction boxes. So, when you think there is an error, drop out of full screen mode and see if the video has a correction box.(4 votes)

## Video transcript

- So I have a function here, h of n, and let's say that it explicitly defines the terms of a sequence. Let me make a little... Let me make a quick table here. We have n, and then we have h of n. When n is equal to one,
h of n is negative 31, minus seven times one minus one, which is going to be... This is just going to be zero, so it's going to be negative 31. When n is equal to two,
it's going to be negative 31 minus seven times two minus one, so two minus one. This is just going to be one, so it's negative 31 minus seven, which is equal to negative 38. When n is equal to three, it's gonna be negative 31 minus seven times three minus one, which is just two, so we're gonna subtract seven twice. It's gonna be negative 31 minus 14, which is equal negative 45. What do we see happening here? We're starting at negative 31, and then we keep subtracting, we keep subtracting negative seven. We keep subtracting
negative seven from that. In fact we subtract negative
seven one less than the term... We subtract negative seven one less times than the term we're dealing with. If we're dealing with the third term we subtract negative seven twice. If we're dealing with the second term we subtract negative seven once. This is all nice, but what I want you to
do now is pause the video and see if you can define
this exact same sequence. The sequence here is you
start at negative 31, and you keep subtracting negative seven, so negative 38, negative 45. The next one is gonna be negative 52, and you go on and on and on. You keep subtracting negative seven. Can we define this sequence in terms of a recursive function? Why don't you have a go at that. Let's try to define it in
terms of a recursive function. Let's just call that g of n, so g of n. In some ways a recursive
function is easier, because you can say okay look. The first term when n is equal to one, if n is equal to one,
let me just write it, If n is equal to one, if n is equal to one, what's g of n gonna be? It's gonna be negative 31, negative 31. And if n, if n is greater than
one and a whole number, so this is gonna be defined
for all positive integers, and whole, and whole number, it's just going to be the previous term, so g of n minus one minus seven, minus seven. We're saying hey if we're just picking an arbitrary term we just have to look at the previous term and then subtract, and
then subtract seven. It all works out nice and easy, because you keep looking at previous, previous, previous terms all the way until you
get to the base case, which is when n is equal to one, and you can build up back from that. You get this exact same sequence. Let's do another example, but let's go the other way around. Here we have a, we have a sequence defined recursively, and I want to create a function that defines a sequence explicitly. Let's think about this. One way to think about it, this sequence, when n is equal to one it starts at 9.6, and then every term is the previous term minus 0.1. The second term is gonna
be the previous term minus 0.1, so it's gonna be 9.5. Then you're gonna go to 9.4. Then you're gonna go to 9.3. We could keep going on and on and on. If we want, we could
make a little table here, and we could say this is n, this is h of n, and you see when n is equal to one, h of n is 9.6. When n is equal to two, we're now in this case over here, it's gonna h of two minus one, so it's gonna be h of one minus 0.1. It's just gonna be this minus 0.1, which is going to be 9.5. When h is three, it's gonna be h of two, h of two minus 0.1, minus 0.1. H of two is right over here. You subtract a tenth you're gonna get 9.4, exactly what we saw over here. Let's see if we can pause the video now and define this... Create a function that
constructs or defines this arithmetic sequence explicitly. Here it was recursively. We wanna define it explicitly. So let's just call it, I don't know, let's just call it f of n. We can say look, it's gonna be 9.6, but we're gonna subtract, we're gonna subtract 0.1 a certain number of times depending on what term
we're talking about. We're gonna subtract 0.1, but how many times are
we gonna subtract it as a function of n? Let's see. If we're talking about the first term we subtract zero times. The second term we subtract one time. The third term we subtract two times. The fourth term we subtract three times. Whatever term we're talking about we subtract that term minus one times. If we're talking about the nth term, we subtracted this
value n minus one times. You can verify that this is going to work. When n is equal to one this term here is going to be zero, so this whole thing is gonna be zero. You get 9.6. When n is equal to two, two minus one, you subtract 0.1 one time. 9.6 minus 0.1 is 9.5. You could keep doing that. You could draw a table, and evaluate these if you want to. The key thing is your'e starting at 9.6 and you're subtracting 0.1 one fewer times than the
term you're looking at. If you're looking at the if n is equal to, this is n is equal to four, well you're gonna subtract 0.1 three times, and you see that. Subtract 0.1 once, subtract 0.1 twice, subtract 0.1 three times.