Final answer:
To find the number of terms required for the sum of the arithmetic sequence to be zero, we set up and solve a quadratic equation using the sum of an arithmetic series formula. The calculation reveals that we need 16 terms for the sum to equal zero.
Step-by-step explanation:
To find out how many terms of the arithmetic sequence 15, 13, 11, 10... must be taken so that the sum of the sequence will be equal to zero, we need to use the formula for the sum of an arithmetic series:
Sn = n/2 (2a + (n - 1)d)
Where Sn is the sum of the first n terms, a is the first term, d is the common difference between the terms, and n is the number of terms.
Given the sequence 15, 13, 11, ..., we have a = 15 and d = -2 (since each subsequent term decreases by 2). We need to find n such that the sum is 0, so setting Sn to zero gives us:
0 = n/2 (2*15 + (n - 1)(-2))
This simplifies to:
0 = n(30 - 2n + 2)
0 = n(32 - 2n)
0 = 32n - 2n2
This is a quadratic equation in the form of an2 + bn + c = 0. To solve for n, we can either factor the equation or use the quadratic formula. Factoring gives us:
n(32 - 2n) = 0
This means n = 0 (which doesn't make sense in this context as we need at least one term) or 32 - 2n = 0 which simplifies to:
n = 16
Therefore, 16 terms of the arithmetic sequence must be taken for the sum to be equal to zero.