$$ \sum _ { i = 1 } ^ { n } ( 3 i - 5 ) = 3430 $$

Question

UpStudy AI · Accepted Answer

To solve the equation:

$$
\sum_{i=1}^{n} (3i - 5) = 3430
$$

Let's break it down step by step.

1. **Express the Sum:**

The sum can be separated into two parts:

$$
   \sum_{i=1}^{n} (3i - 5) = 3\sum_{i=1}^{n} i - 5\sum_{i=1}^{n} 1
   $$

2. **Simplify the Sums:**

- The sum of the first $$ n $$ natural numbers is:

$$
     \sum_{i=1}^{n} i = \frac{n(n + 1)}{2}
     $$

- The sum of $$ n $$ ones is:

$$
     \sum_{i=1}^{n} 1 = n
     $$

Plugging these into the original expression:

$$
   3 \cdot \frac{n(n + 1)}{2} - 5n = 3430
   $$

3. **Solve for $$ n $$:**

Multiply both sides by 2 to eliminate the fraction:

$$
   3n(n + 1) - 10n = 6860
   $$

Expand and rearrange:

$$
   3n^2 + 3n - 10n - 6860 = 0 \
   3n^2 - 7n - 6860 = 0
   $$

4. **Use the Quadratic Formula:**

The quadratic equation is:

$$
   3n^2 - 7n - 6860 = 0
   $$

Using the quadratic formula $$ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$:

- $$ a = 3 $$
   - $$ b = -7 $$
   - $$ c = -6860 $$

Calculate the discriminant:

$$
   \Delta = (-7)^2 - 4 \cdot 3 \cdot (-6860) = 49 + 82320 = 82369
   $$

$$
   \sqrt{82369} = 287
   $$

So,

$$
   n = \frac{7 \pm 287}{6}
   $$

We discard the negative solution as $$ n $$ must be positive:

$$
   n = \frac{7 + 287}{6} = \frac{294}{6} = 49
   $$

**Answer:** $$ n = 49 $$
$$ n = 49 $$

\( \sum _ { i = 1 } ^ { n } ( 3 i - 5 ) = 3430 \)