MATH SOLVE

10 months ago

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

Accepted Solution

A:

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

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