Find n for a series for which a649-14-01-00-00_files/i0160000.jpg = 30, d = -4, and sn = -210.

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

Ok here we go ..there are plenty of ways to solve such math problems but we will go for the smartest and easiest ..5x+2y=7  -2x+6y=9we rearrange them into the following ..by multiplying the 1st equation by 2 it becomes 10x + 4y = 14we rearrange it and so it becomes 10x = 14 - 4yas for the second equation we multiply it by 5 and rearrange it so it becomes as to the following ..-10x + 30y = 45rearranging so 10x = 30y -45now 10x equals both 14 - 4y and 30y -45so   14 - 4y = 30y -4531 = 26yy= 31/26 = 1.19 or around 1.2so by now we can calculate x easily as  -2x+6y=9so 2x = 6y -9 = 6 * 31/26 - 9 = -1.84  then x = - 0.92 or around -0.9best luck!