To find the first negative term in the arithmetic progression (AP) 2000, 1990, 1980, 1970 ..., we need two key components of the arithmetic progression: the first term and the common difference.
1. First, identify the first term and the common difference in the given sequence.
The first term (a) of the AP is 2000.
The common difference (d) of the AP can be calculated by subtracting any term from the term that directly follows it. So, we subtract the second term from the first term:
1990 - 2000 = -10.
So, the common difference is -10.
2. We know that the nth term (T) of an AP can be found using the formula: T = a + (n-1)d. To find the first negative term, we have to set T to be less than 0 and solve for n.
2000 + (n-1)(-10) < 0
It simplifies as: 2000 - 10n + 10 < 0
Simplifying further, we get: 10n > 2010
Which gives n > 201.01
3. As we are dealing with terms, n should be rounded up (since we can't have, say, 1.5 of a term in a sequence). Therefore, the term number should be the next integer greater than 201.02 which is 202.
4. We then substitute n = 202 into the nth term formula T = to calculate the 202nd term, which is the first negative term.
T202 = 2000 + (202 - 1)(-10) = 2000 - 2010 = -10
So, the first negative term is the 202nd term and the term itself has a value of -10.