Question

6-5. given the sequence 7, 11, 15, 19, ...
a. what kind of sequence is it?
b. define the sequence explicitly.
c. is 109 a term of the sequence? if so, which term is it?

Explanation:

Step1: Identify sequence type

Check the common difference between consecutive terms:
$11-7=4$, $15-11=4$, $19-15=4$. The difference is constant.

Step2: Write explicit formula

Use arithmetic sequence formula $a_n = a_1 + (n-1)d$, where $a_1=7$, $d=4$:
$a_n = 7 + (n-1) \times 4 = 4n + 3$

Step3: Check if 109 is a term

Set $a_n=109$, solve for $n$:
$4n + 3 = 109$
$4n = 109 - 3 = 106$
$n = \frac{106}{4} = 26.5$

Answer:

a. It is an arithmetic sequence.
b. The explicit formula is $a_n = 4n + 3$ (where $n$ is a positive integer, $n \geq 1$)
c. 109 is not a term of the sequence, since the term number $n$ must be a positive integer, and we got $n=26.5$ which is not an integer.