the first term of the arithmetic sequence is -37 and the second term is-39 what is the smallest positive term of the sequence?
To find the common difference (d) of an arithmetic sequence, we subtract any two consecutive terms. Let's subtract the second term (-39) from the first term (-37):
-37 - (-39) = -37 + 39 = 2
Therefore, the common difference (d) is 2.
To find the nth term of an arithmetic sequence, we use the formula:
nth term = first term + (n - 1) * common difference
Since we are looking for the smallest positive term, we need to find the value of n that gives a positive result.
Let's substitute the known values into the formula:
nth term = -37 + (n - 1) * 2
We want the nth term to be positive, so we set the equation equal to zero and solve for n:
-37 + (n - 1) * 2 > 0
Simplifying the inequality:
-37 + 2n - 2 > 0
2n - 39 > 0
2n > 39
n > 39/2
n > 19.5
Since n must be an integer, the smallest value of n that satisfies the inequality is 20.
Now, we can substitute n = 20 into the nth term formula to find the smallest positive term:
nth term = -37 + (20 - 1) * 2
nth term = -37 + 19 * 2
nth term = -37 + 38
nth term = 1
Therefore, the smallest positive term in the arithmetic sequence is 1.
To find the smallest positive term of the arithmetic sequence, we need to determine the common difference first. The common difference (d) can be calculated by subtracting the second term from the first term:
d = second term - first term
= (-39) - (-37)
= -39 + 37
= -2
Now that we know the common difference is -2, we can find the terms of the sequence by adding the common difference repeatedly to the first term. However, since we are looking for the smallest positive term, we need to find the first positive term of the sequence.
To calculate the first positive term, we need to find the smallest non-negative value for n in the equation:
n(d) + a ≥ 0
Where:
- n represents the position of the term in the sequence
- d is the common difference
- a is the first term
Substituting the given values:
n(-2) - 37 ≥ 0
Simplifying the equation:
-2n - 37 ≥ 0
-2n ≥ 37
n ≤ -37/2
Since n represents the position of the term, it must be a positive integer. The largest value of n that satisfies the inequality n ≤ -37/2 is 18. Therefore, the smallest positive term can be found by substituting this value into the equation for the nth term:
n = 18
term = a + (n-1)(d)
= -37 + (18-1)(-2)
= -37 + 17(-2)
= -37 - 34
= -71
So, the smallest positive term of the arithmetic sequence is -71.