The number of terms of the arithmetic progression (AP) 18, 16, 14,... that must be taken for their sum to be zero is 19.
Explanation
Here's how to determine the number of terms:
We are given an arithmetic progression (AP) with the first term, a = 18, and a common difference, d = 16 - 18 = -2. We want to find the number of terms, n, such that the sum of these n terms, Sn, is zero.
The formula for the sum of the first n terms of an AP is:
Sn = (n/2) [2a + (n - 1)d]
In this case, Sn = 0, a = 18, and d = -2. Plugging these values into the formula:
0 = (n/2) [2(18) + (n - 1)(-2)]
0 = (n/2) [36 - 2n + 2]
0 = (n/2) [38 - 2n]
Multiplying both sides by 2, we get:
0 = n(38 - 2n)
This gives us two possible solutions for n:
- n = 0
- 38 - 2n = 0 => 2n = 38 => n = 19
Since we're looking for a number of terms to take, n=0 is not applicable in this instance, as we need to take at least one term. Therefore, the correct answer is n = 19.
According to the provided reference, the answer is indeed 19.
