Given that −6, − 2 1
2
, . . . , 71 is a linear
sequence, Calculate the number of terms in the
sequence.
A. 21
B. 22
C. 23
D. 24
To find the number of terms in a linear sequence, we need to first find the common difference (d) between consecutive terms. We can do this by subtracting the second term from the first term:
d = (-21/2) - (-6) = (-21/2) + 12/2 = -9/2
Now we can find the number of terms (n) in the sequence using the formula:
a_n = a_1 + (n-1)d
where a_1 is the first term, a_n is the last term, and d is the common difference.
We know that a_1 = -6 and a_n = 71, so we can substitute these values into the formula along with d = -9/2:
71 = -6 + (n-1)(-9/2)
Multiplying both sides by -2/9 gives:
-142/9 = -12 + (n-1)
Adding 12 to both sides gives:
102/9 = n-1
Simplifying gives:
n = 121/9
This is not a whole number, so we need to round up to the nearest integer since the sequence contains a finite number of terms. Therefore, the number of terms is:
n = 14
But we need to add one more term to include the second term (-21/2) that we left out when calculating the common difference. Therefore, the total number of terms in the sequence is:
n = 14 + 1 = 15
Answer: There are 15 terms in the sequence, which is not one of the options given in the answer choices. Therefore, this question may contain an error.
To find the number of terms in the sequence, we need to determine how many terms are there starting from -6 and ending at 71, with a common difference of 2.
We can use the formula for the nth term of an arithmetic sequence:
aₙ = a₁ + (n-1)d
In this case, a₁ = -6, d = 2, and aₙ = 71. Let's substitute these values into the formula and solve for n:
71 = -6 + (n-1)2
71 = -6 + 2n - 2
71 + 6 - 2 = 2n
77 = 2n
Dividing both sides by 2:
n = 77 / 2
n = 38.5
Since the number of terms should be a whole number, we can round n up to the nearest whole number. Therefore, the number of terms in the sequence is 39.
None of the options given matches the actual answer.