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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.
• 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!
• this is clearer approach:
15-6(100-1)
=15-594
=-579
• 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?
• 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.
• Why is it 15-99x6 instead of 15-100x6?
(1 vote)
• 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).
• how do you know when to use a recursive or explicit formula for a math problem? Do they give you a general rule of when to use it or not use it? What I think is that since the previous terms weren't given, Sal couldn't use a recursive formula since he would have to know all the terms. However, I'm not sure that this is correct.
• You use whichever is easier with that specific equation. For example, like in the video, finding 99 terms before getting the value you're looking for would be very time consuming and very annoying. So, an explicit formula works better.
(1 vote)
• The setup equation is 15-(n-1)* 6
How did you know to multiply by *6?
I would have automatically made the mistake of putting -6, in the set up equation charting the sequences lol.
(1 vote)
• technically it would be the same thing, your way would just yield an = 15 + (n-1) * -6. So in fact, you would not have made a mistake at all.
• how did he change it 2 a positive 6?
(1 vote)
• What do you mean? What minute?

As far as I see he never changed from -6 to +6.
(1 vote)
• So what is the nth term of the following sequence
1,2,4,7,11,16,22
(1 vote)
• You will notice that the sequence is adding +1,+2,+3 and so on. Can you pick up from here?
(1 vote)
• the seventh number in the square sequence ___
(1 vote)
• The sum of 3 consecutive odd number 315, what are the numbers?
(1 vote)
• lets X be the odd number, the next consecutive odd (skip an even number) would be x+2. And the next consecutive odd would be x+2+2. Let put them in table form.

x
x+2
x+2+2

the sum of x+(x+2)+(x+2+2)=315
solve for x will give you the first odd number, then you can find the next two.
(1 vote)