Question

1. the sequence ... is defined explicitly as ...

what is the recursive form of this sequence?
a. ( a_1 = -5, a_n = -3a_{n - 1} )
b. ( a_1 = -5, a_n = 10a_{n - 1} )
c. ( a_1 = -5, a_n = a_{n - 1} - 3 )
d. ( a_1 = -5, a_n = a_{n - 1} + 3 )

1. the sequence ( a_n ) is defined by ( a_1 = 4 ) and ( a_n = a_{n - 1} + 3 ). what is the explicit form of the sequence if written in a simplified format without parentheses?
2. the table below records the median salary at united construction over a period of 4 years.

year	salary
2010	$36,000
2011	$38,000
2012	$40,000
2013	$43,000

what is the average annual increase in the median salary from 2009 to 2013?
a. $2,000
b. $8,000
c. $28,000

Explanation:

Question 6

Step1: Find first term $a_1$

Assume explicit form is $a_n=3n-8$ (matches options).
$a_1=3(1)-8=-5$

Step2: Find recursive relation

$a_n=3n-8$, $a_{n-1}=3(n-1)-8=3n-11$
$a_n - a_{n-1}=(3n-8)-(3n-11)=3$
So $a_n=a_{n-1}+3$

Question 7

Step1: Identify sequence type

It is arithmetic sequence: $a_1=4$, $d=3$

Step2: Apply explicit formula

Arithmetic sequence formula: $a_n=a_1+(n-1)d$
$a_n=4+(n-1)3=4+3n-3=3n+1$

Question 8

Step1: Calculate total salary increase

Total increase = $43000-35000=8000$

Step2: Calculate number of years

Years passed: $2013-2009=4$

Step3: Compute average annual increase

$\text{Average increase}=\frac{8000}{4}=2000$

Answer:

1. D. $a_1=-5, a_n=a_{n-1}+3$
2. $a_n=3n+1$
3. A. $\$2,000$