Precalculus Unit 5 Exam ☹️ (copy)

What operations happen in an arithmetic sequence?

Add or subtract (linear)

What operations happen in a geometric sequence

Multiply or divide (exponential)

“A_n=2n+3” What is the constant rate, and what letter is represented?

2 or D

“A_n=2n+3” What kind of sequence is this? What form is it written in?

Arithmetic; explicit

“A₁ = 1; A_n = A_n-1 + 3”

What is d? What kind of equation is this? What form is this? How do you turn this into explicit

3=d, constant rate

Arithmetic

Recursive

Take d • n + or - whatever needed to make d • n = A₁

Recursive Geometric

A_n = A_n-1 • r

What is r in geometric sequences?

The ratio or constant rate

Explicit geometric

A_n=A₁( r )^n-1

Required set up for Recurrence Relation

F_n=A(F_n-1)+B(F_n-2)

What form do you turn the recurrence relation into first

x²+Ax+B=0

What do you do when you have the recurrence relation as a quadratic?

Solve for roots: X₁ and X₂

Equation for real roots characteristic polynomial

F_n=ax₁ⁿ+bx₂ⁿ

What should your final answer be after solving a characteristic polynomial system?

Explicit form

What equation is used if you are given:

S_{n =} ⁿΣ_i=#(A_n where form is dn + #)

What will n represent in that found equation? How do you find n?

(First + Last Terms) * n

2

N represents the # of terms. You find it by subtracting First - Last + 1.

What equation is used if you are given:

S_{n =} ⁿΣ_i=# (A_n where form is A₁*(r)^n-1)

S_n = [first term - a](1-rⁿ)

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1-r

What special conditions can be used for a simple formula to find the sum of a geometric series?

if |r| < 1, n —> ∞ (end term is infinity)

What is the second, simplified formula for sum of a geometric series only used in some cases?

S_∞=^∞Σ_n=1(a(r)^n-1) =

a

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1-r

Convert to explicit

a₁=7

a_n = a_n–1 – 3

a_n=-3n +10

Convert to explicit

a₁ = -41

a_n=a_n-1–11

a_n=-11n-30

Convert to recursive

a_n=-8n-29

a₁=-37

a_n=a_n-1 – 8

Convert to recursive

a_n= -15n +34

a₁=19

a_n=a_n-1-15

Geometric mean equation

^nth√ΠA_n

Arithmetic mean equation

ΣA_n

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_n

Harmonic mean equation

n

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1

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a_n

If r in a geometric series > 1, growth —>

∞

If r in a geometric series is < 0, —>

alternating

If r in a geometric series is < 1, and > -1, but is not 0, decay —>

0

If r in a geometric series is 0, growth —>

0