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Math  /  Algebra

Question4. Given the following sequence, which statement below is true? (1 Point) −18,−9,0,9,18,………648-18,-9,0,9,18, \ldots \ldots \ldots 648 The ratio of the sequence is 9 The common difference of the sequence is 9 The sequence above has 70 terms The sum of the sequence is 23620

Studdy Solution

STEP 1

1. The sequence given is arithmetic.
2. The sequence starts at −18-18 and ends at 648648.
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)=9d = -9 - (-18) = 9

STEP 5

Use the formula for the nn-th term of an arithmetic sequence, an=a1+(n−1)⋅da_n = a_1 + (n-1) \cdot d, to find the number of terms. Given an=648a_n = 648, a1=−18a_1 = -18, and d=9d = 9:
648=−18+(n−1)⋅9648 = -18 + (n-1) \cdot 9
Solve for nn:
648+18=(n−1)⋅9648 + 18 = (n-1) \cdot 9
666=(n−1)⋅9666 = (n-1) \cdot 9
n−1=6669n-1 = \frac{666}{9}
n−1=74n-1 = 74
n=75n = 75

STEP 6

Calculate the sum of the sequence using the formula for the sum of an arithmetic sequence, Sn=n2⋅(a1+an)S_n = \frac{n}{2} \cdot (a_1 + a_n):
S75=752⋅(−18+648)S_{75} = \frac{75}{2} \cdot (-18 + 648)
S75=752⋅630S_{75} = \frac{75}{2} \cdot 630
S75=75⋅315S_{75} = 75 \cdot 315
S75=23625S_{75} = 23625

STEP 7

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."

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