Question

algebra 1
unit 6 test review
name_________
date_______ per______
arithmetic rules
explicit: $f(n)=f(1)+d(n - 1)$
recursive: $f(n)=f(n - 1)+d, f(1)=$ ______
geometric rules
explicit: $f(n)=f(1)(r)^{n-1}$
recursive: $f(n)=f(n - 1)r, f(1)=$ ______

1. write the first 5 terms of the geometric sequence defined by rule $f(n)=f(n - 1)\cdot6, f(1)=3$
2. write the first 5 terms of the arithmetic sequence defined by rule: $f(n)=f(n - 1)-4, f(1)=33$
3. write an explicit and recursive rule for the arithmetic sequence 92, 72, 52, 32, ...

$f(1)=$
$d=$
explicit: $f(n)=$
recursive: $f(n)=$
$f(1)=$

1. write an explicit and recursive rule for the geometric sequence 2, -6, 18, -54, ...

$f(1)=$
$r=$
explicit: $f(n)=$
recursive: $f(n)=$
$f(1)=$

Explanation:

Problem 1: Find first 5 geometric terms

Step1: Identify initial term

$f(1) = 3$

Step2: Calculate 2nd term

$f(2) = f(1) \cdot 6 = 3 \cdot 6 = 18$

Step3: Calculate 3rd term

$f(3) = f(2) \cdot 6 = 18 \cdot 6 = 108$

Step4: Calculate 4th term

$f(4) = f(3) \cdot 6 = 108 \cdot 6 = 648$

Step5: Calculate 5th term

$f(5) = f(4) \cdot 6 = 648 \cdot 6 = 3888$

Problem 2: Find first 5 arithmetic terms

Step1: Identify initial term

$f(1) = 33$

Step2: Calculate 2nd term

$f(2) = f(1) - 4 = 33 - 4 = 29$

Step3: Calculate 3rd term

$f(3) = f(2) - 4 = 29 - 4 = 25$

Step4: Calculate 4th term

$f(4) = f(3) - 4 = 25 - 4 = 21$

Step5: Calculate 5th term

$f(5) = f(4) - 4 = 21 - 4 = 17$

Problem 3: Arithmetic sequence rules

Step1: Identify initial term

$f(1) = 92$

Step2: Find common difference $d$

$d = 72 - 92 = -20$

Step3: Write explicit formula

$f(n) = 92 + (-20)(n-1) = 112 - 20n$

Step4: Write recursive formula

$f(n) = f(n-1) - 20$, with $f(1)=92$

Problem 4: Geometric sequence rules

Step1: Identify initial term

$f(1) = 2$

Step2: Find common ratio $r$

$r = \frac{-6}{2} = -3$

Step3: Write explicit formula

$f(n) = 2(-3)^{n-1}$

Step4: Write recursive formula

$f(n) = f(n-1) \cdot (-3)$, with $f(1)=2$

Answer:

1. First 5 terms: $3, 18, 108, 648, 3888$
2. First 5 terms: $33, 29, 25, 21, 17$
3. $f(1)=92$, $d=-20$

Explicit: $f(n)=112-20n$
Recursive: $f(n)=f(n-1)-20$, $f(1)=92$

1. $f(1)=2$, $r=-3$

Explicit: $f(n)=2(-3)^{n-1}$
Recursive: $f(n)=f(n-1)\cdot(-3)$, $f(1)=2$