Precalc 8.1-8.3: Sequences and Series

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25 Terms

1
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nth term of an arithmetic sequence
aₙ = a₁ + (n-1)d
2
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sum of a finite arithmetic sequence
n/2(a₁ + aₙ)
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recursion formula
aₙ = a sub (n-1) + d
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geometric sequence formula
aₙ = a₁r^n-1
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d =
common difference
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r =
common ratio
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aₙ =
nth term
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n =
term number
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sum of a finite geometric sequence
Sₙ **=** ∑n, i=1, a₁r^i-1 **=** a₁(1-rⁿ)/(1-r)
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sum of an infinite geometric series
If |r| is less than 1, Sₙ **=** ∑∞, i=0, a₁r^i **=** a₁/(1-r)
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For sums of **finite** geometric sequences , if the index begins at i=0, you must…
adjust the formula so the sigma becomes n=1
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n! =
n(n-1)(n-2)…\*2\*1 and 0! = 1
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∑caₙ =
c∑aₙ (c is a constant)
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∑c (n is stop point on top of ∑)
cn
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∑(aₙ + bₙ) =
∑aₙ + ∑bₙ
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∑(aₙ - bₙ) =
∑aₙ - ∑bₙ
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Write recursion formula for the sequence

15, 7, 8, -1, 9, -10…
aₙ = a sub(n-2) - a sub(n-1), n ≥ 3
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How many terms are in ∑ 100, n=51, n
50 (do limit-start+1)
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Number of terms in a sequence
first term - last term + 1
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Find the sum of the integers from 40 to 80
Sₙ = 41/2(40 + 80)

Sₙ = 2460

(THERE ARE 41 TERMS FROM 40 TO 80)
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3!8!/4!4! =
3!8x7x6x5x4!/4x3!x4!

Cancel out factorials

8x7x6x5/4
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Formula for 2, 4, 8, 16, 32, 64
aₙ = 2ⁿ
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How to solve annuity problems

1. Find a₁ and r using A=P(1+r/n)^nt, a₁ is the last payment which is only compounded once, r is value inside the parentheses
2. Set up a sum of a finite geometric sequence equation with n = total number of months compounded
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Write the recursive and explicit formula for the sequence

11, 101, 1001, 10001…
Recursion: aₙ = 10(a sub(n-1)) -9

Explicit: aₙ = 10ⁿ +1
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Find an expression for the nth partial sum of

aₙ = (1/n+1) - (1/n+2)
Telescoping series (only first and last terms remain)

Sₙ = (1/2-1/3) + (1/3-1/4) + … + (1/(n+1)) - (1/(n+2))

Cancel out all terms besides first and last

Sₙ = 1/2 - 1/(n+2)

Sₙ = n/(2n+4)