The sum of the first 4 terms of an arithmetic series is -8 and the sum of the first 5 terms is 85. Determine the first term and the common difference.

Steps to Solve

1. Write the formula for the sum of an arithmetic series

The sum $S_n$ of the first $n$ terms of an arithmetic series is given by the formula:

$$S_n = \frac{n}{2}[2a + (n-1)d]$$ where $a$ is the first term and $d$ is the common difference.

2. Apply the formula for the sum of the first 4 terms

We are given that the sum of the first 4 terms is -8, so $S_4 = -8$. Using the formula:

$$S_4 = \frac{4}{2}[2a + (4-1)d] = -8$$

Simplify the expression:

$$2(2a + 3d) = -8$$

Divide both sides by 2:

$$2a + 3d = -4$$

3. Apply the formula for the sum of the first 5 terms

We are given that the sum of the first 5 terms is 85, so $S_5 = 85$. Using the formula:

$$S_5 = \frac{5}{2}[2a + (5-1)d] = 85$$

Simplify the expression:

$$\frac{5}{2}(2a + 4d) = 85$$

Multiply both sides by $\frac{2}{5}$:

$$2a + 4d = 34$$

4. Solve the system of equations

Now we have a system of two equations with two variables:

$$2a + 3d = -4$$ $$2a + 4d = 34$$

Subtract the first equation from the second equation to eliminate $a$:

$$(2a + 4d) - (2a + 3d) = 34 - (-4)$$

$$d = 38$$

5. Solve for $a$

Substitute the value of $d$ into the first equation to solve for $a$:

$$2a + 3(38) = -4$$

$$2a + 114 = -4$$

$$2a = -118$$

$$a = -59$$