Math Snap

STEP 1

1. The sequence given is arithmetic.
2. The sequence starts at $-18$ and ends at $648$.
3. We need to determine which statement about the sequence is true.

STEP 2

1. Determine the type of sequence.
2. Calculate the common difference.
3. Calculate the number of terms in the sequence.
4. Calculate the sum of the sequence.
5. Evaluate the given statements.

STEP 3

Identify the sequence type by observing the pattern. The sequence increases by a constant amount, indicating it is an arithmetic sequence.

STEP 4

Calculate the common difference by subtracting the first term from the second term:
$$d = -9 - (-18) = 9$$

STEP 5

Use the formula for the $n$-th term of an arithmetic sequence, $a_n = a_1 + (n-1) \cdot d$, to find the number of terms. Given $a_n = 648$, $a_1 = -18$, and $d = 9$:
$$648 = -18 + (n-1) \cdot 9$$ Solve for $n$:
$$648 + 18 = (n-1) \cdot 9$$ $$666 = (n-1) \cdot 9$$ $$n-1 = \frac{666}{9}$$ $$n-1 = 74$$ $$n = 75$$

STEP 6

Calculate the sum of the sequence using the formula for the sum of an arithmetic sequence, $S_n = \frac{n}{2} \cdot (a_1 + a_n)$:
$$S_{75} = \frac{75}{2} \cdot (-18 + 648)$$ $$S_{75} = \frac{75}{2} \cdot 630$$ $$S_{75} = 75 \cdot 315$$ $$S_{75} = 23625$$

Evaluate the given statements:
- The ratio of the sequence is 9: FALSE, as this is an arithmetic sequence, not geometric.
- The common difference of the sequence is 9: TRUE.
- The sequence above has 70 terms: FALSE, it has 75 terms.
- The sum of the sequence is 23620: FALSE, the sum is 23625.
The true statement is:
"The common difference of the sequence is 9."