I want to verify that this is correct:
An arithmetic sequence begins, 116, 109, 102
• Determine whether -480 belongs to this sequence, if it does, what is its term number?
-480 = 116 + (n - 1)(-7)
n - 1 = 85.142. . .
So, that means -480 does not belong to this sequence, right? Thank you for any helpful replies!
Correct!
Or in modular arithmetic:
116-(-480) mod 7 = 1 ≠0
OK thank you!
To verify whether -480 belongs to the arithmetic sequence, we can substitute -480 into the formula for an arithmetic sequence and solve for the term number, denoted as "n".
The formula for an arithmetic sequence is:
$s_n = a + (n - 1)d$
Where:
$s_n$ is the term number "n",
$a$ is the first term of the sequence,
and $d$ is the common difference between consecutive terms.
In this specific arithmetic sequence, the first term, $a$, is 116, and the common difference, $d$, is -7.
Substituting these values into the formula, we have:
-480 = 116 + (n - 1)(-7)
Now, let's solve for n:
-480 = 116 - 7n + 7
-480 - 116 + 7 = -7n
-480 - 109 = -7n
-480 - 109 + 109 = -7n + 7n
-480 - 109 = 0
The equation -480 - 109 = 0 does not hold true, meaning that there is no value of n that satisfies this equation. Therefore, -480 does not belong to this arithmetic sequence.
In conclusion, your understanding is correct. -480 does not belong to this arithmetic sequence, and there is no corresponding term number for it.