How many terms of the arithmetic sequence 88, 85, 82, . . . appear before the number -17 appears?
in your sequence, a = 88, d = -3
so you want a + (n-1)d = -17
88 + (n-1)(-3) = -17
88 - 3n + 3 = -17
-3n = -108
n = 36
state your conclusion
To find the number of terms of the arithmetic sequence that appear before the number -17 appears, we need to find the common difference and the pattern of the sequence.
The given arithmetic sequence is 88, 85, 82, ...
To find the common difference (d), we can subtract any two consecutive terms in the sequence. Let's subtract the second term from the first term:
85 - 88 = -3
From this calculation, we can see that the common difference is -3.
Now, we can determine the pattern of the sequence. Since the common difference is negative, the sequence is decreasing.
We want to find the number of terms before the number -17 appears. Let's set up an equation to solve for this:
88 + (-3)(n - 1) = -17
Here, n represents the number of terms.
To solve for n, we need to simplify the equation and solve for n:
88 - 3n + 3 = -17
Combine like terms:
91 - 3n = -17
Subtract 91 from both sides of the equation:
-3n = -17 - 91
Simplify:
-3n = -108
Divide both sides by -3:
n = -108 / -3
n = 36
Therefore, there are 36 terms of the arithmetic sequence that appear before the number -17.
To find the number of terms before -17 appears in the arithmetic sequence, we need to determine the position of -17 in the sequence.
The given arithmetic sequence is: 88, 85, 82, ...
We can calculate the position of -17 by finding the common difference and then using the general formula for the nth term of an arithmetic sequence.
The common difference (d) can be determined by subtracting any two consecutive terms.
In this case, we can subtract the second term (85) from the first term (88):
d = 85 - 88 = -3
Now, we can use the formula for the nth term of an arithmetic sequence:
aₙ = a₁ + (n - 1)d
Where:
aₙ = nth term of the sequence
a₁ = first term of the sequence
d = common difference
n = position of the term in the sequence
We want to find the position of the term -17, so we can set up the equation:
-17 = 88 + (n - 1)(-3)
Simplifying the equation, we get:
-17 = 88 - 3n + 3
Now, let's solve for n:
-17 - 88 + 3 = -3n
-102 = -3n
Divide both sides of the equation by -3 to solve for n:
n = -102 / -3
n = 34
Therefore, the number -17 appears at the 34th position in the arithmetic sequence.
To answer the original question, we need to determine the number of terms before the 34th position. Since the sequence starts at 88 and ends at -17, we can subtract 34 from the total number of terms in the sequence:
Total number of terms = last term - first term + 1 = -17 - 88 + 1 = -104
Number of terms before -17 appears = Total number of terms - 34 = -104 - 34 = -138
Therefore, there are 138 terms before the number -17 appears in the given arithmetic sequence.