Question

practice #2-7
$a_n=a_1+(n-1)(d)$
first number common difference
write the explicit form for the sequence $a_1 = 25$ $d = -4$
write the explicit form for the sequence 4, 6, 8, 10, ...
write the first 4 terms of the sequence.
a. $a_n = 125 + (n - 1)(-25)$
b. $a_n = -25 + (n - 1)(25)$
c. $a_n = 16 + (n - 1)(4)$
d. $a_n = -4 + (n - 1)(-10)$
you can use the explicit form to find ____ in a sequence. to find an unknown value, write the number you are looking for where $a_n$ is. replace $a_1$ with the __ number of the sequence. replace the $d$ with the ________.
-4, 28, 32, ...
find $a_7$

Explanation:

Step1: Identify explicit formula parts

The explicit formula for an arithmetic sequence is $a_n = a_1 + (n-1)d$, where $a_1$ is the first term, $d$ is the common difference, and $n$ is the term number.

Step2: Solve first explicit form

Given $a_1=25$, $d=-4$. Substitute into formula:
$a_n = 25 + (n-1)(-4)$
Simplify: $a_n = 25 -4(n-1) = 29 -4n$

Step3: Solve second explicit form

Sequence: 4,6,8,10,...; $a_1=4$, $d=6-4=2$. Substitute:
$a_n = 4 + (n-1)(2)$
Simplify: $a_n = 4 +2n-2 = 2n+2$

Step4: Match explicit form option

Check each option:

• A. $a_n=125+(n-1)(-25)$: $a_1=125$, $d=-25$
• B. $a_n=-25+(n-1)(25)$: $a_1=-25$, $d=25$
• C. $a_n=16+(n-1)(4)$: $a_1=16$, $d=4$
• D. $a_n=-4+(n-1)(-10)$: $a_1=-4$, $d=-10$

(Note: No matching prompt given, but this is the breakdown)

Step5: Fill the blanks

1. "You can use the explicit form to find any term in a sequence."
2. "Replace $a_1$ with the first number of the sequence. Replace the $d$ with the common difference."

Step6: Find $a_7$ for -4,28,32,...

First, find $d$: $28-(-4)=32$, $32-28=4$ (corrected $d=4$, first term $a_1=-4$)
Use formula: $a_7 = -4 + (7-1)(4)$
Calculate: $a_7 = -4 + 24 = 20$

Step7: Write first 4 terms (no explicit sequence given, assume general: if using option C: $a_1=16$, $a_2=20$, $a_3=24$, $a_4=28$; if using first problem's sequence: 25, 21, 17, 13)

Answer:

1. Explicit form for $a_1=25, d=-4$: $\boldsymbol{a_n = 25 + (n-1)(-4)}$ or $\boldsymbol{a_n=29-4n}$
2. Explicit form for 4,6,8,10,...: $\boldsymbol{a_n = 4 + (n-1)(2)}$ or $\boldsymbol{a_n=2n+2}$
3. Filled blanks:

• You can use the explicit form to find any term in a sequence.
• Replace $a_1$ with the first number of the sequence. Replace the $d$ with the common difference.

1. $a_7$ for -4,28,32,...: $\boldsymbol{20}$
2. First 4 terms (for $a_1=25, d=-4$): $\boldsymbol{25, 21, 17, 13}$