We have:
a₁ = 30
d = -4
Sn = -210
n = ?
The formula to find an is:
an = a₁ + (n - 1)d
Substituting the given numbers:
an = 30 + (n - 1)(-4) = -4n + 34
The formula for Sn is:
Sn = n/2 · (a₁ + an)
Substituting the formula found for an, we get:
Sn = n/2 · (30 + (-4n + 34)
= n/2 · (30 - 4n + 34)
= n/2 · (64 - 4n)
= 32n - 2n²
We know that Sn = -210, therefore we need to solve the equation:
32n - 2n² = -210
2n² - 32n - 210 = 0
n² - 16n - 105 = 0
Which gives you two solutions:
n₁ = -5 (not acceptable)
n₂ = 21
Therefore, the correct answer is n = 21