nth Term of an Arithmetic Sequence

an = a₁ + (n-1)d

3-5 Arithmetic Sequences as Linear Functions

5, 10, 15, 20, ... is a sequence! ! ! ! !! ! ! Itʼs terms are: ! ! Itʼs common difference, d, is 5, because the terms increase by 5. (Itʼs called a common difference because to find it you subtract the previous term from the next term...like 15-10=5 and 20-15=5.)

If the difference between terms is the same, or common, then itʼs an arithmetic sequence.

Ex. 1C Are these arithmetic sequences?

-26, -22, -18, -14,... ____

1, 4, 9, 25, ... ____

Ex. 2A Find the next four terms: 9.5, 11.0, 12.5, 14.0, ____, ____, ____, ____

Investigate: 3, 6, 9, 12, 15, ...

! *What is d? ___! *Name the first term ___! *Notice the second term a₂ is the first term ___ ! ! ! ! ! plus the common difference ___, or... a₂ = a₁+ d!! *So the third term ___ is the first term ___ plus the common difference ___ how many times? ___ So... ! ! ! ! ! ! a₃ = a₁+ 2d

*How would you describe a₄?! ! ! ! ! ! _________ *How would you describe a₅?! ! ! ! ! ! _________ *How would you describe a₆?! ! ! ! ! ! _________

*What pattern do you see? ___________________________________________ *What would a₁₀₀ be? ____ *If an is the nth term, what would an be? __________

(Everything above is the derivation of the formula.)

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+5 +5+5

Ex. 3E Given any sequence, you could be asked to do the following. Letʼs take for example the sequence: ! ! 1, 10, 19, 28,...

Write an equation for the nth term.an = a₁ + (n-1)d *you know a₁ = ___

! ! ! ! ! ! ! ! *you know d = ___! ! ! ! ! ! ! ! *plug them in and simplify! ! ! ! an = 1 + (n-1)9! ! ! ! an = 1 + 9n - 9! ! ! ! an = 9n - 8! ! ! *notice the form, one variable term ! ! ! ! ! ! ! ! ! and one constant term**If asked to write in function notation... ! follow the same process, but write the final answer as f(n) = 9n -8.

Find the 12th term of the sequence. (Use formula you just found...DONʼT write out the sequence to the 12th term.)

Graph the first five terms. The points are (n, an)

! ! ! ! Term 1 is 1 ! ( 1, 1)! ! ! ! Term 2 is 10! ! ( 2, 10)! ! ! ! Term 3 is 19 ! ( 3, 19)! ! ! ! Term 4 is ___! ( , )! ! ! ! Term 5 is ___! ( , )! ! ! ! ! ! ! ! *Sketch a graph above and plot these. " " " " " " " " " " DONʼT connect.

Which term of the sequence is 172?! ! ! ! an = a₁ + (n-1)d ! *Start with formula, plug in knowns, solve! ! ! 172 = 1 + (n-1)9! ! for unknown. (You could have started with! ! ! 172 = 1 + 9n - 9! ! the formula you found earlier (9n-8), but ! ! ! 172 = 9n - 8! ! ! sometimes they jump straight to asking you ! ! ! +8 +8! ! ! this question, without having already asked ! ! ! 180 = 9n! ! ! for the formula for the nth term, so we are ! ! ! ! 9 9!! ! ! just going back to the basic formula.)! ! ! 20 = n! ! ! So 172 is the 20th term.

*Note that sometimes youʼll be asked to find these same things, but instead of being given the sequence, youʼll be given, say....that a₁₂ is 22 and d=-5. You could use this in the formula to find a₁ and then anything else that is asked. (Because you know n=12, an=22, and d=-5.)

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Biblical Integration: Sequence Creator

As we study and begin to recognize sequences and patterns in creation, the evidence of a Creator is blinding. "For ever since the creation of the world His invisible nature and attributes, that is, His eternal power and divinity, have been made intelligible and clearly discernible in and through the things that have been made (His handiworks). So [men] are without excuse [altogether without any defense or justification]," Romans 1:20

Problems for Practice: 7, 9, 11, 15, 19, 27 (reduced from video) Complete on notebook paper or the back of these printed notes, in notes section of binder. Check with the back of the book. Place a colored check by correct answers. Correct incorrect answers to the right. Use Hotmath if you need help.

Also complete the following Skills Review: none

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