$$ \sum _ { p = 5 } ^ { m } ( 2 - 3 p ) = - 7085 $$

Question

UpStudy AI · Accepted Answer

To solve the equation:

$$
\sum_{p=5}^{m} (2 - 3p) = -7085
$$

we need to find the value of $$ m $$.

### Step 1: Understand the Summation

The summation is an arithmetic series where each term is given by $$ 2 - 3p $$ for $$ p $$ ranging from 5 to $$ m $$.

### Step 2: Express the Sum of the Series

The sum of an arithmetic series can be calculated using the formula:

$$
S_n = \frac{n}{2} \times (\text{first term} + \text{last term})
$$

Where:
- $$ n = m - 4 $$ (since the series starts at $$ p = 5 $$)
- The first term $$ a = 2 - 3 \times 5 = -13 $$
- The last term $$ l = 2 - 3m $$

So, the sum becomes:

$$
S_n = \frac{m - 4}{2} \times (-13 + (2 - 3m)) = -7085
$$

### Step 3: Simplify the Equation

$$
\frac{m - 4}{2} \times (-11 - 3m) = -7085
$$

Multiply both sides by 2:

$$
(m - 4)(-11 - 3m) = -14170
$$

Expand the left side:

$$
-3m^2 + m + 44 = -14170
$$

Bring all terms to one side:

$$
-3m^2 + m + 14214 = 0
$$

Multiply by -1 to make it easier to solve:

$$
3m^2 - m - 14214 = 0
$$

### Step 4: Solve the Quadratic Equation

Use the quadratic formula:

$$
m = \frac{1 \pm \sqrt{1 + 4 \times 3 \times 14214}}{2 \times 3}
$$

Calculate the discriminant:

$$
\sqrt{1 + 170568} = \sqrt{170569} = 413
$$

So,

$$
m = \frac{1 \pm 413}{6}
$$

This gives two solutions:

1. $$ m = \frac{414}{6} = 69 $$
2. $$ m = \frac{-412}{6} \approx -68.67 $$ (Discard this since $$ m $$ must be a positive integer)

### Conclusion

The value of $$ m $$ that satisfies the equation is:

$$
m = 69
$$
$$ m = 69 $$

\( \sum _ { p = 5 } ^ { m } ( 2 - 3 p ) = - 7085 \)