Precalc 8.1-8.3: Sequences and Series

nth term of an arithmetic sequence

aₙ = a₁ + (n-1)d

sum of a finite arithmetic sequence

n/2(a₁ + aₙ)

recursion formula

aₙ = a sub (n-1) + d

geometric sequence formula

aₙ = a₁r^n-1

d =

common difference

r =

common ratio

aₙ =

nth term

n =

term number

sum of a finite geometric sequence

Sₙ **=** ∑n, i=1, a₁r^i-1 **=** a₁(1-rⁿ)/(1-r)

sum of an infinite geometric series

If |r| is less than 1, Sₙ **=** ∑∞, i=0, a₁r^i **=** a₁/(1-r)

For sums of **finite** geometric sequences , if the index begins at i=0, you must…

adjust the formula so the sigma becomes n=1

n! =

n(n-1)(n-2)…\*2\*1 and 0! = 1

∑caₙ =

c∑aₙ (c is a constant)

∑c (n is stop point on top of ∑)

cn

∑(aₙ + bₙ) =

∑aₙ + ∑bₙ

∑(aₙ - bₙ) =

∑aₙ - ∑bₙ

Write recursion formula for the sequence

15, 7, 8, -1, 9, -10…

aₙ = a sub(n-2) - a sub(n-1), n ≥ 3

How many terms are in ∑ 100, n=51, n

50 (do limit-start+1)

Number of terms in a sequence

first term - last term + 1

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)

3!8!/4!4! =

3!8x7x6x5x4!/4x3!x4!

Cancel out factorials

8x7x6x5/4

Formula for 2, 4, 8, 16, 32, 64

aₙ = 2ⁿ

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

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

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)