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## Algebra 1

### Course: Algebra 1 > Unit 9

Lesson 2: Constructing arithmetic sequences- Recursive formulas for arithmetic sequences
- Recursive formulas for arithmetic sequences
- Recursive formulas for arithmetic sequences
- Explicit formulas for arithmetic sequences
- Explicit formulas for arithmetic sequences
- Explicit formulas for arithmetic sequences
- Arithmetic sequence problem
- Converting recursive & explicit forms of arithmetic sequences
- Converting recursive & explicit forms of arithmetic sequences
- Converting recursive & explicit forms of arithmetic sequences
- Arithmetic sequences review

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# Arithmetic sequence problem

Sal finds the 100th term in the sequence 15, 9, 3, -3... Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

- To find the sum for arithmetic sequence, sn= n(n+1)/2, it is shown (n+1)/2, can be replaced with the average of nth term and first term. How do we understand that we should not replace the "n" outside the bracket should not be replaced with nth term too.(5 votes)
- Confusingly, "n" IS the nth term in this particular sequence!

The ( n + 1 ) represents the sum of the last term (n) and the first term (1).

Dividing by 2 gives us their average.

Then we multiply that by the number of terms (n).

Hope this makes things clearer!(6 votes)

- Finding the 100th term (or any term that's not given) is pretty straightfoward with an explicit(ly defined) equation. But how do you do it with a recursive(ly defined) equation?

eg with the recursive equation for this video's example: a(100)=a("subscript" 100-1) - 6

As in, you don't have the 99th term's value so how do you find it so you can then subtract 6 from it and get the 100th term's value?(3 votes)- You want to get the analytic form (= explicitly defined) for your recursive sequence. One, kind of hand-wavey way to do it would be to calculate some amount of the first terms, try to spot the pattern and define the analytic expression.

Another way to do it, presuming it's of the appropriate form, would be to use the first-order linear recurrence equation.

If you have a recursively defined sequence a_n = c*a_(n-1) + d, and you're given the first term a_0, then the sequence explicitly defined is:

a_n = a_0 * c^n + d * (c^n - 1) / (c - 1).

Notice that if c = 1, then you have just a regular arithmetic sequence.(1 vote)

- Why is it 15-99x6 instead of 15-100x6?(2 votes)
- We're asked to seek the value of the 100th term (aka the 99th term after term # 1). We're given the first term = 15, therefore we need to find the value of the term that is 99 terms after 15. The arithmetic formula shows this by a+(n-1)d where a= the first term

(15), n= # of terms in the series (100) and d = the common difference (-6).(6 votes)

- What is the difference between sequence and series?...(2 votes)
- Sequence: Particular Format of Elements

Series:**Sum**of the elements in a sequence.

E.G : Sequence would be 1,2,3,4...

E.G : Series would be 1+2+3+4...

As you see, the Sequence*helps*the series. The Sequence*shows*the numbers, while the Series*adds*the numbers.

Hope this helped! 😁(5 votes)

- Sal i need help on this problem −3= t/−15

(2 votes)- Multiply 15 to both sides -15(-3)=(t/-15)15 then you get -15(-3)=t then you multiply the -15(-3) so then the answer is 45=t on the left-hand side the 15's
**canceled**out which you were left with**T**.(2 votes)

- Is there ever the -1th term in a number sequence/pattern?(2 votes)
- Yes, but they are rarely mentioned as it has little to no significance.(2 votes)

- I just wrote it as b(100)=15-6(100-1) and easily got -579. Do I need to do all of those extra steps he did?(1 vote)
- Most of Sal's extra steps are to find the equation. You need the equation to quickly get to the 100th term.(3 votes)

- isnt the equatiion supposed to be n=a1+(n-1)d?

how did he get 15-(n-1)d?(2 votes)- 15 is written as a1 and the - sign is given instead of + is because the number further decreases(1 vote)

- how do you find the 261 term?(1 vote)
- That would be about 15 - (260)6 which would be -1545(2 votes)

- An arithmetic progression consists of 26 terms. Given the first term is 2 and the sum of the

last 8 terms is 532. Find the 15th term of the progression. So how do I solve this?(1 vote)- We have an arithmetic progression

𝐴 = {𝑎(1), 𝑎(2), 𝑎(3), ..., 𝑎(26)}

We also have

𝑎(19) + 𝑎(20) + 𝑎(21) + ... + 𝑎(26) = 532

Since 𝑎(𝑛 + 1) = 𝑎(𝑛) + 𝑑, we can write this sum as

𝑎(19) + 𝑎(19) + 𝑑 + 𝑎(19) + 2𝑑 + 𝑎(19) + 3𝑑 + ... + 𝑎(19) + 7𝑑 =

= 8 ∙ 𝑎(19) + (1 + 2 + 3 + ... + 7)𝑑 =

= 8 ∙ 𝑎(19) + 28𝑑 = 532 ⇒

⇒ 𝑎(19) = (532 − 28𝑑)∕8 = 66.5 − 3.5𝑑

𝑎(𝑛) = 𝑎(1) + (𝑛 − 1)𝑑 ⇒ 𝑎(19) = 𝑎(1) + 18𝑑

𝑎(1) = 2 ⇒ 𝑎(19) = 2 + 18𝑑

Thereby, 66.5 − 3.5𝑑 = 2 + 18𝑑 ⇒

⇒ 𝑑 = (66.5 − 2)∕(18 + 3.5) = 3

𝑎(15) = 𝑎(1) + 14𝑑 = 2 + 14 ∙ 3 = 44(2 votes)

## Video transcript

- [Instructor] We are
asked what is the value of the 100th term in this sequence, and the first term is 15, then nine, then three, then negative three. So let's write it like this in a table. So if we have the term, just so we have things straight, and then we have the value, and then we have the value of the term. I'll do a nice little table here. So our first term we saw is 15. Our second term is nine. Our third term is three. I'm just really copying this down, but I'm making sure we associate
it with the right term. And then our fourth term, our fourth term is negative three. And they wanna ask, they want us to figure
out what the 100th term of this sequence is going to be. So let's see what's happening here, if we can discern some type of pattern. So we went from the first
term to the second term, what happened? 15 to nine, looks like
we went down by six. It's always good to
think about just how much the numbers changed by. That's always the
simplest type of pattern. So we went down by six, we subtracted six. Then to go from nine to three, well we subtracted six again. We subtracted six again. And then to go from
three to negative three, well we, we subtracted six again. We subtracted six again. So it looks like every term, you subtract six. So the second term is going to be six less than the first term. The third term is going to be 12 minus from the first term, or six subtracted twice. So in the third term, you subtract a six twice. In the fourth term, you subtract six three times. So whatever term you're looking at, you subtract six one less
than that many times. Let me write this down just so, notice when your first term, you have 15 and you don't
subtract six at all, or you could say you
subtract six zero times. So you could say this is 15 minus six times or let me
write it better this way, minus zero times six. That's what that first
term is right there. What's the second term? This is 15, it's just we just subtracted six once, or you could say minus one times six. Or you could say plus
one times negative six, either way, we're
subtracting the six once. Now what's happening here? This is 15, this is 15 minus two times negative six, or sorry minus two times six, minus two times six. We're subtracting a six twice. What's the fourth term? This is 15 minus, we're subtracting the six
three times from the 15, so minus three times six. So if you see the pattern here, when our term, when we
have our fourth term, we have the term minus one right there, the fourth term we have a three, the third term we have a two, the second term we have a one. So if we had the nth term, if we just had the nth term here, what's this going to be? It's going to be 15 minus, you see it's going to
be n minus one right here, right when n is four,
n minus one is three. When n is three, n minus one is two. When n is two, n minus one is one. When n is one, n minus one is zero. So we're going to have, this term right here is n minus one, so minus n minus one times six. So if you wanna figure out the
100th term of this sequence, I didn't even have to write
it in this general term, you could just look at this pattern. It's going to be, and I'll do it in pink, the 100th term in our sequence, I'll continue our table
down, is gonna be what? It's going to be 15 minus 100 minus one, which is 99, times six, right? I just followed the pattern. One, you had a zero here,
two, you had a one here, three, you had a two here, 100, you're gonna have a 99 here. So let's just calculate what this is. What's 99 times six? So 99 times six, actually you could do this in your head. You could say that's going to be six less than 100 times six, which is 600, and six less is 594. But if you didn't wanna do it that way, you just do it the old-fashioned way. Six times nine is 54, carry the five. Nine times six or six times nine is 54. 54 plus five is 594. So this right here is 594, and then to figure out what 15, so we wanna figure out, we wanna figure out what 15 minus 594 is, and this can sometimes be confusing, but the way I always
process this in my head is I say that this is the exact same thing as the negative of 594 minus 15. And if you don't believe me, distribute out this negative sign. Negative one times 594 is negative 594, negative one times
negative 15 is positive 15. So these two statements are equivalent. This is much easier for
my brain to understand. So what's 594 minus 15? You should do it, we
could do this in our head. 594 minus 14 would be 580, and then 580 minus one more would be 579. So that right there is 579, and then we have this negative
sign sitting out there. So our the 100th term in our sequence will be negative 579.