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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!
• 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.
(1 vote)
• Why is it 15-99x6 instead of 15-100x6?
• 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).
• What is the difference between sequence and series?...
• 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! 😁
• Sal i need help on this problem −3= t/−15
• 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.
• Is there ever the -1th term in a number sequence/pattern?
• Yes, but they are rarely mentioned as it has little to no significance.
• 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.
• isnt the equatiion supposed to be n=a1+(n-1)d?
how did he get 15-(n-1)d?
• 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
• 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