Question

1. write the explicit arithmetic equation.

a. 2, 5, 8, 11....
what is the 99th term?
is 365 a term in the sequence?
why or why not?
b. 50, 32, 14, -4....
what is the 26th term?
is -789 a term in the sequence?
why or why not?

Explanation:

Part a: Sequence 2, 5, 8, 11,...

Step1: Find common difference $d$

$d = 5 - 2 = 3$

Step2: Write explicit formula

The explicit formula for an arithmetic sequence is $a_n = a_1 + (n-1)d$, where $a_1=2$, $d=3$.
$a_n = 2 + (n-1) \times 3 = 3n - 1$

Step3: Calculate 99th term

Substitute $n=99$ into $a_n = 3n - 1$.
$a_{99} = 3(99) - 1 = 297 - 1 = 296$

Step4: Check if 365 is a term

Set $3n - 1 = 365$, solve for $n$.
$3n = 366 \implies n = 122$
Since $n=122$ is a positive integer, 365 is a term.

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Part b: Sequence 50, 32, 14, -4,...

Step1: Find common difference $d$

$d = 32 - 50 = -18$

Step2: Write explicit formula

Use $a_n = a_1 + (n-1)d$, where $a_1=50$, $d=-18$.
$a_n = 50 + (n-1)(-18) = 68 - 18n$

Step3: Calculate 26th term

Substitute $n=26$ into $a_n = 68 - 18n$.
$a_{26} = 68 - 18(26) = 68 - 468 = -400$

Step4: Check if -789 is a term

Set $68 - 18n = -789$, solve for $n$.
$-18n = -857 \implies n = \frac{857}{18} \approx 47.61$
Since $n$ is not a positive integer, -789 is not a term.

Answer:

Part a

1. Explicit equation: $a_n = 3n - 1$
2. 99th term: $296$
3. 365 is a term, because solving $3n-1=365$ gives a positive integer $n=122$.

Part b

1. Explicit equation: $a_n = 68 - 18n$
2. 26th term: $-400$
3. -789 is not a term, because solving $68-18n=-789$ gives a non-integer $n=\frac{857}{18}$.